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JOURNAL OFFUNCTIONAL ANALYSIS 9, 121-180 (1972)

Hypercontractive Semigroups

and Two Dimensional Self-Coupled Bose Fields

BARRY SIMON AND RAPHAEL H~EGH-KROHN*+

Department of Mathematics, Princeton University, Princeton, New Jersey

Received November 30, 1970

We present an abstract perturbation theory for operators of the form HO + V obeying four properties: (1) H, is a positive self-adjoint operator on L2(M, JL) with p a probability meaaure so that e- tHo is a contraction on L’ for each t > 0; (2) e-rHo is a bounded map of Le to L4 for some T; (3) V E L’(M, p) for some p > 2; (4) ectv EL’ for all t > 0. We then show that spatially cutoff Bose fields in two-dimensional space-time fit into this framework. Finally, we discuss some details of two-dimensional Bose fields in the abstract including coupling constant analyticity in the spatially cutoff case.

Contents

1. Introduction . . . . . . . . . . . . . . . 122

2. A Perturbation Theory for Hypercontractive Semigroups

A. Hypercontractive Semigroups . . . .

B. Lm Perturbations . . . . . . . .

C. Almost Semi-Bounded Lp Perturbations (p > 2)

D. Analytic Perturbation Theory . . _ . . . .

E. Extensions . . . . . . . . .

. 125

. 125

127

. 130

137

. 138

139

140

3. Hypercontractive Semigroups in Fock Space . . . . .

A. Coherent Vectors . . . . . . . . .

B. Quantized Operators as Generators of Hypercontractive Semi- groups . . . , . . . . . . . . . . . .

C. WickPolynomials in Two-Dimensional Space-Time . . . .

4. Spatially Cutoff Two-Dimensional Bose Field Theories . . . .

A. Lower Bounds and Self-Adjointness . . . .

B. The Vacuum Vector for the Spatially Cutoff Theory . . .

C. Coupling Constant Dependence . . . . . .

145

150

156

156

159

169

5. Applications to Non-Relatavistic Quantum Mechanics . . . . . . 176

* On leave from Mathematisk Institut, Oslo University, Norway. + Research partially sponsored by the USAFOSR under contract AF49(638) 1545.

121 0 1972 by Academic Press, Inc.

580/9/a-r

122 SIMON AND HOEGH-KROHN

I. INTRODUCTION

It is our purpose here to give a comprehensive treatment of the mathematical properties of spatially cutoff self-coupled Bose field Hamiltonians in two-dimensional space-time; every attempt has been made to keep this paper comprehensible to the mathematically sophisticated reader who has only a minimal background in quantum field theory.

We view the work we describe here as a synthesis of two lines of development both of which have their roots in the pioneering study of Nelson [45]. On the one hand, Glimm-Jaffe [19-221 and Rosen [47, 481 using Nelson’s estimates and techniques suggested by Jaffe’s study of the strongly cutoff (finite number of degrees of freedom) theory [33] have developed an impressive amount of information for (~4)s and (c$%)~ theories. On the other hand, Segal [52, 531 using essentially an abstraction of the properties Nelson isolated has provided a streamlined proof of semiboundedness, suggested a way of defining the Hamiltonian as a formal sum, and made a series of elegant comments. In Section 2, we will continue Segal’s abstract study and obtain a series of stronger results, the most important of which is the essential self-adjointness of the operator sum H, + V. This provides a proof of essential self-adjointness for the V = Jg(x) : P(~(x)) : dx case considerably simpler than the existing proofs of Glimm-Jaffe and Rosen [ 19,471.

Since we are able to develop the results in Section 2 without ever introducing an ultraviolet cutoff, we are motivated to attempt to prove that two-dimensional Boson self-coupled field theories have all the properties needed to apply the methods of Section 2 by an approach which uses a minimal number of cutoffs. We do this in Section 3 by borrowing a technique from quantum optics: The formalism of coherent vectors [38]. The results of Section 3 have on the whole, already been proven (explicitly or implicitly) by Nelson [45], Glimm [18], Jaffe-Glimm [19], and Segal [53]. Their proofs always involve putting in approximations and the use of Fatou’s lemma. We have attempted, on the other hand, to deal as directly as possible with the objects of interest themselves. It is our feeling that any trend towards fewer cutoffs in the proofs must be regarded as a healthy trend.

In Section 4, we will consider special aspects of the theory of self- coupled Boson fields in two dimensional space-time. Some of these results have already been obtained by Glimm and Jaffe [20-221 in the p4 case, by Rosen [47, 481 in the vzn case and by Segal [53] in the abstract setting. In our study of the vacuum vector, we will present

HYPERCONTRACTIVE SEMIGROUPS 123

some new results which are natural in the LP setting, and we will present a streamlined proof of uniqueness of the vacuum vector for the spatially cutoff theory [21]. We also present a discussion of analyticity in the coupling constant and of asymptotic perturbation series for thiz spatially cutoff theory.

We should like to conclude this introduction by attempting a comparison of our techniques (which originate with Segal) and those of Glimm-Jaffe.l First, we should remark that we only deal with one part of the results of Glimm-Jaffe. Their work can be divided into two parts; those parts which require “higher order estimates,“2 i.e., inequalities of the form A n < B” (n, m not both 1) and those which rely on first order estimates only. This division is not always well- defined; for example, the Glimm-Jaffe proof of essential self- adjointness [19, 201 1 re ies on higher order estimates, while Rosen’s proof [47] needs no higher order estimates. Results that depend critically on higher order estimates cannot be obtained by the methods of this paper, but we remark that a great deal of the theory only needs first order estimates. In any event, the next step in an abstract development along the lines laid down by Segal and extended by us in this paper should be to attempt to translate the higher order estimates into the abstract framework.

We have been unable to prove the essential self-adjointness of the Cannon-Jaffe locally correct Lorentz generator [g], although we feel the modified theory of Section 2E (Theorem 2.26) might be applicable. In any event, once one knows H, + HI is essentially self-adjoint (which we can prove), the Cannon-Jaffe proof of essential self- adjointness only depends on one elementary quadratic estimate and on the Kato-Rellich theory of regular perturbations.

What then are the main technical simplifications we have accom- plished ? First, by using fewer cutoffs, our proofs in Section 3 are simpler looking in many instances although these proofs are essentially the same as the highly cutoff proofs. Secondly, we have found an elementary Fock space derivation of Mehler’s formula [42] avoiding special formulas for Hermite functions [27] or the theory of Brownian motion [13]. Most crucially, we have eliminated all need of path

1 The remainder of this introduction is not needed for the bulk of this paper and is intended primarily for readers familiar with the details of the work of Glimm and Jaffe.

2 The most famous such higher order estimate is the quadratic estimate

Ho2 + H,* < a[H(g) + b12

for pp field theories [20].


integration and the Feynman-Kac formula. Once one understands our proofs, one can see that they are in some simple sense the skeletons of the Glimm-Jaffe proofs. We have replaced the need for Feynman- Kac with the Trotter formula and have replaced the positivity of the path space measure with the fact that e+*o is a contr’action on the LY spaces. It is well known [44] that once one has established the countable additivity of the path space measure, the Feynman-Kac formula is nothing but the Trotter product formula. The contractivity of e-lHo follows from the positivity of the kernel of ~~~0,~ which is essentially all that is needed for the construction of a cylinder set measure on the paths. The place where path integration techniques are stronger than the Trotter product formula plus contractivity is where the full measure is needed rather than the measure on just the cylinder sets; establishing the countable additivity of the full measure is the deep part of the theory of path space integration and this fact is apparently not needed in the applications made thus far in field theory.

Finally, by developing the theory in an abstract setting, we gain all the advantages of abstraction: Our proofs have been pared down to the essentials; the abstract setting suggests certain questions which aren’t obvious in the concrete situation but whose answer would be illu- minating; and it is possible to extend the theory to other cases of interest. We will, in fact, raise certain questions about the cutoff vacuum which are natural in this setting-questions for which a positive answer would have important consequences. And it is clear that our methods will handle some infinite Wick series V = Jg(x) x:,“=O a,: v%(x): dx. We have not attempted to isolate precise conditions on the a, which will allow us to carry the theory through; we remark that severe fall offof a, as n + co is needed; for example, the an’s given by cos x = C,“=,, a,xn don’t fall off rapidly enough to get e-Vt E L1. We also note that our techniques can be modified to handle interactions of the form JIz ,< ~ g(x) +exp n&x)+-- dx for E sufficiently small; this theory will be the subject of a separate note by one of us [28].

In summary then, we have isolated the crucial mathematical ideas behind a segment of the important work of Glimm and Jaffe by elaborating on the techniques suggested by Segal. The result seems to us to be a theory not only of great physical interest, but also of considerable mathematical beauty.

3 And it is equivalent once one knows Ho1 = 1.


2. A PERTURBATION THEORY FOR HYPERCONTRACTIVE SEMIGROUPS

In this section, we consider the following abstract setting: We have an operator H,, on L2(M, p) with TV a probability measure so that e--tHo (t > 0) is a contraction on all theLp spaces and e-THo is a bounded map from L2 to L4 for some T > 0. We are interested in defining Ho + V as a nice operators on L2 for a set of perturbations V which are unbounded multiplication operators.

It was Nelson [45] who first emphasized the crucial nature of the contractive property of e--IHo in field theory models. Recently, Segal [52, 531 considered the precise problem we consider in this section. The reader should view Sections 2B and the first part of 2C as a systematization of Segal’s work. We go on to extend Segal’s results in two ways: (a) We prove resolvant convergence of H,, + V, to H, + V when V, converges to V in a certain sense; (b) we prove H, + V(x) is an analytic family in the sense of Kato [35], where x -+ V(z) is analytic in a certain sense we make precise in Section 2D.

A. Hypercontractive Semigroups

Let M be a fixed measure space with p probability measure (positive measure of total mass 1) on M.

DEFINITION. Let H,, > 0 be a self-adjoint operator on L2(M, p). We say e+flo (t > 0) is a hypercontractive semigroup if and only if:

(4 ee6Ho is a contraction on L1(M, p) for all t > 0.

(b) For some T (which we denote T(H,,)), eeTHo is a bounded map from L2(M, p) to L4(M, p).

When (a) but not necessarily (b) holds, we say e-6Ho is contractive. For the reader’s convience, we first recall that condition (a) is

implied if e-lx0 has a “probability kernel”; that is:

PROPOSITION 2.1. Let e-tHo be positivity preserving (i.e., f >, 0 pointwise implies e-6Hof >, 0 pointwise) and suppose e--1fW&, = 9, where Q,, is the function Q,(x) = 1 f or all x. Then ebHo is a contraction on WM, P).

Proof. Since e-6H 0 is positivity preserving, we need only prove Ij e+%/ II1 < I/ $ (Ii for # 3 0. But since II, 3 0 and e-6Ho# > 0:

I/ eP”* Ill = (Q, , e-tHo#) = (e-tHoQ, , #>

= (Q, > 1cI> = II *Ill * Q.E.D.


Various interpolation theorems imply that hypercontractive semigroups have additional contractive properties. The Reisz-Thorin theorem [14] coupled with duality (e-lHo is its own dual) implies:

PROPOSITION 2.2. Let e+Ho be a hypercontractive semigroup. Then

(a> ertHo is a contraction on Lp(M, p) for all 1 < p < 00 and all t > 0.

(b) Let 1 < a < b < a be given. Then there is a Ta,b and a C so that

for all t > T,,, and all a < p, q < 6.

Proof. (a) By duality, e+*o is a contraction on L” and by Reisz-Thorin on all Lp.

(b) s ince e-tHo is a bounded map from L2 to L4 and from L” to L”, it follows from Reisz-Thorin that it is a bounded map from Lp to L2P ( p > 2). Thus, e- %THo is a bounded map from L2 to L2n. Find n so that a,, = 2n/2n - 1 < a < b < 2” = b, . Then e--2nfH0 is a bounded map from L Uo to Lb0 (using the above and its dual). Since

;; lg II # II* if P G q and eetHo is a contraction on each Lp, 0 ows. Q.E.D.

In what follows, we will need control over estHo for t small. This we can obtain through the use of the Stein interpolation theorem4 which we recall says [56]:

PROPOSITION 2.3 (Stein interpolation theorem). Let H(x) be a family f p f ji t 1 o ma s rom m e inear combinations of characteristicfunctions of sets of jinite measure on some measure space to its dual. Suppose

(a) H(z) is analytic in the strip 0 < Re z < T, continuous on the boundaries.

(b) Jf Re 2 = 0, II W+h lip, d G II SL II4 and ;f Re z = T, II f@MJ Ilp, d CI II # lIPI .’ Then for t real, 0 < t < T (s = t/T),

II fw # IlP$ G G II II, IL

4 It is a pleasure to thank Dr. C. Fefferman for valuable discussions about this theorem.

6 The original theorem allows growth of C,,,, in the Im z direction.


with p,l = sprl + (1 - s) p,l; qrl = $4:’ + (1 - 8) $1,

log ct = s log c, + (1 - s) log c,

Proof. See Ref. [56].

PROPOSITION 2.4 (Segal [53]). Let e-tHo be a hypercontractive semigroup and let p(t, 6) be defined by

p-1 = p(t, 8)-l + t6.

Then given 1 < a < b < CO, there is a 6 and C (both depending on a and 6) so that:

for all a < p < b and all t < T(H,J.

Proof. Consider first the case a = b = 2. Let f(z) = emsHo be defined for 0 < Re z < T. f ( ) x is analytic in the strip, continuous on the boundary; llf(4~llz < II #II2 for Re .Z = 0 and Ilf(4#l14 < C 11 $J [I2 for Re z = T (where C is the norm of e-IX0 as a map of L2 to L4).

Thus, by the Stein interpolation theorem, the result follows with 6 = 1/4T. The general a, b case follows from the a = b = 2 case, the contractive nature of e-tHo on L” and the Reisz-Thorin theorem.

Q.E.D.

Finally, by a remark of Stein, we have:

PROPOSITION 2.5. Let e-tHo be a contractive semigroup. If

then e-=Ho is a contraction of Lp to IF.

Proof. See Stein [57] for this result which follows from applying the Stein interpolation theorem to exp HO(veiot) = f (t). Q.E.D.

B. L” Perturbations

Following Segal [53], we first consider the case of V EL” (V is not necessarily real), and derive estimates that only depend on

I/ e-tv /I1 = s ectReV dp = (11 e-tm-lv lo,)“.


The simplest estimate is the following:

PROPOSITION 2.6 (Segal [53]). Let V eLrn and let e-tHo be a contractive semigroup. Let 00 3 p > q 3 1 with q-l = p-l -t m-l. Then:

Proof. Consider A, = (e-6Ho/ne-1Yln)n. By Holder’s inequality Ij e-tvln# ~I+I+~-+-I,-~ < 11 e@‘ln llAIIL /I # lip . By repeated use of this sort of inequality and the contractive nature of e-tHoJn on all J?, we see

that II A,# II4 < II e+ln IIE, II 4 Ilp = II @‘IL II $ Ilp . By the Trotter product formula [59, lo], A,# -% e-l(Ha+Y)+ for say tj EL”. Since the A,# are in a fixed Lq sphere, they have a weak L’J limit point which must be the unique L2 limit point. Thus 11 e-t(HO+V)# /Iq < /j e-tv Ilrn jl# lip .

Q.E.D.

Proposition 2.6 is very weak in that even the identity is a contraction from Lp to Lq if p > q. But using the strong inequalities for hypercontractive e&Ho contained in Proposition 2.3, we can conclude strong inequalities for eWt(Ho+J’). First:

PROPOSITION 2.7 (Segal [53]). Let V EL” and let e--tHo be a hypercontractive semigroup. Let 1 < p < CO. Then there are a C and an m (dependent on p and H,, but independent of V and t) so that

II e- t(HO+VV IUp G C”(ll e-” Mtll # I& .

Proof. As in the proof of Proposition 2.6, we need only prove for t fixed and all n sufficiently large that we have

IIW tV’ne-tHo’n)n I) j12, ,( Ct /I e-” Ilm)tlI J/J iI9 .

This follows if we can prove

!I@- tv’ne-tHo’n) Q!J lip < Ct’“(ll e-” lIm)t’nll * II9

for ra sufficiently large (t fixed). By Proposition 2.4, we can find 6 (dependent on p) so that

II e-tHo’la# Il(p-l-ts,nj-l < Pn II 1cI II9 .


By Holder’s inequality,

Since (/ e-tvin jlnlts = (11 e-” lI1,s)“ln the proof is completed. Q.E.D.

In the case p = 2, the above result can be restated in the form:

COROLLARY 2.8 (Segal [53]). Let H,, be a hypercontractive semigroup and let E(V) = inf spec(H, + V) for V EL”, real. Then, there is a D and an m (independent of V) so that

E(V) > -D - log /I e-” Ilrn .

Such lower boundedness theorems in the field theory context originated in the work of Nelson [45] and were refined by Glimm [18]; see also Federbush [15]. The exact form of Corollary 2.6 is stated in Segal’s note [52] and a very elegant (alternate) proof is sketched based on the following lemma:

LEMMA 2.9 (Segal). If // ecAj2ecBe-A12 11 < C for A and B semibounded operators on a Hilbert space and if A + B is essentially self- adjoint on D(A) n D(B), then /I e-(A+B) (/ < C.

Proof. Segal’s proof is based on monotonicity of the square-root on positive operators and the Trotter formula. Alternately, we can prove this by noting that Lowner’s theorem [41] implies log X < log Y if 0 < X < Y. Thus, // e-Aj2e-Be-A/2 11 < C which implies e-B < Ce+A also implies -B < log C + A or -(A + B) < log C. Q.E.D.

Notes. (1) Our proof using the monotonicity of log, proves A + B > -log C as a statement of quadratic forms even if A + B is not self-adjoint. (2) For the reader’s convenience we point out that the monotonicity of logarithm follows directly from the Herglotz representation formula for the logarithm:

j

0

log x = (1 -kYX) -m (1 +Y2)(Y -qdy.

One can strengthen Proposition 2.7 to the form:

PROPOSITION 2.10. Let e-txo be a hypercontractive semigroup. Let 1 < p, q < co. Then there is an S, an E and an m so that for all V EL”:


Proof. Suppose 2q > p > q first. As in Proposition 2.4, pick C, 6 so that for all r, q < r 6 (2p-1 - q-l)-l,

for all t < T. Let m = (2S6)-1, where S is given by q-l = p-l + S&/2. As usual, we need only prove

ll(e- vS’ne-HoS’n)n 1c, jlz, < C jj e-‘” (Iml! * IIn

for all n sufficiently large. Pick n so S/n < T. Then

By Holder’s inequality

II e- vs’ne-H~s’Y ll~P-l-S6,2n)-1 d CSIn II e--YS’nl12n,S6 II # /IQ.

Iterating this argument, we see

IICe- VS’ne-HoS’n)n # ll,~-l-ss,z < Cs II e-VS’nlI~n,SG II # II4

so that

II e--(Ho+v)s# llp < Cs II em”’ Ml # II* +

For arbitrary p > q, we iterate the above argument. Q.E.D.

In summary then, we have seen L” perturbations yield semigroups with properties only slightly weaker than hypercontractive semigroups: Instead of being contractions from L” to LP all 1 < p < co, the perturbed semigroups are bounded from Lp to Lp for 1 < p < co (for V EL”, they are bounded also from L” to L” but the bounds depend on 11 VII-,,) and e- T(Ho+V) is still a bounded map for L2 to L4 for some T. More crucially, the bounds only depend on powers of the quantities J e- Vs dp. This suggests the results of this section should extend to a wider class of perturbations.

C. Almost Semibounded Lp Perturbations (p > 2)

Again following Segal, we extend some of our results from Section 2B to a class of non-L” potentials. Not surprisingly, we will deal with potentials V so that J e&V dp < co. Thus while V is not necessarily bounded below, it can only get large and negative on very


small sets (~((x j V(X) < -C}) < D,eTCt for all t), i.e., it is “almost bounded below.” We will see that the condition V E Lp for some p > 2 enters naturally.

The idea of Segal is to use DuHamel’s formula to prove e-l(Ho+Vm) converges to a semigroup whose generator he defines to be the (formal) sum H, + V (the V, are Lp approximations to Y which he in L”). We follow Segal’s argument [53] for defining II,, + V, but our proof that H + V so defined is actually essentially self-adjoint on D(H,,) n D(V) differs slightly from Segal’s proof. If one bears in mind that two-dimensional field theories fit into the framework of almost semibounded Lp perturbations and also recalls that spatially cutoff field theoretic Hamiltonians have been proven essentially self-adjoint as operator sums by Glimm-Jaffe [20] and Rosen [47], then the general essential self-adjointness result is not really very surprising.

We first recall DuHamel’s formula:

LEMMA 2.11. Let A be positive operator on a Hilbertspace H. Let B, C be bounded self-adjoint operators. Then

e-(t--uL4+B)(C _ B) e--21L4+C) due

Proof. Let $ E D(A). Thenf(t) = e- t(a+B)# is the unique solution off(O) = #;f’(t) = -(A + B)f(t). Let

&t) = e--tL4+C) # + 1: e-(t-u)(a+B)(C - B) e-u(A+C)# due

An elementary computation shows g’(t) = -(A + B) g(t) so

g(t) = f(t)- Q.E.D.

To emphasize where hypercontractivity as opposed to contractivity enters our arguments, let us first state a result that only require e-lrro to be contractive.

PROPOSITION 2.12 (Segal). Suppose V E Lp (some p > 2) and e-tv E L1 all t > 0. Let

an = W), if I WI B n, = 71, if 1 V(x)1 > n.

Let e-tHo be a con&active semigroup and let H, = H, + V, . Then for


any co>r>s>l, with s-l > r-l + p-l and any fixed t > 0 ectHm converges as maps of Lr + LS (in the norm topology) unaformly on compact subsets of [0, a). The operators e-lH so defined are bounded mups of L’ to L” for all r > s and obey

II e-9 Ils < II e-tvl/mll * IIT

where s-l = r-l + m-l. For any r > s > 1, and any z,b EL’, e-lH$ -+ e-““t+b in the weak L” topology.

Proof (Segal). Pick real numbers j, K so r > i > K > s and K-i = j-l + p-l. By Proposition 2.4, 11 e+@ofV+ /Ii < C, 11 4 llr for all n and all 0 < u < t for some constant C, independent of n. Since V, -+ V in LP, I/( V, - V,) e--“(Ho+V+ Ilk -+ 0 uniformly for $ in the unit ball of L* (by Holder’s inequality). Finally, the e-(t-u)(Ho+V+J are uniformly bounded as maps of Lk to L” so the first statement is proven by using DuHamel’s formula.

Now let r and s (r > s > 1) be arbitrary. We have /I e-%/ /Is < I/ e-tV lIm II+ /IT so by the weak compactness of balls in LS, e-lH# has a weak L”-limit point. As in the proof of Proposition 2.6, this limit point k ;oiL+5 which lies in a suitable LS ball for the /I e-tv lIm inequality

Q.E.D.

Until we obtain e-tH from some Lp to itself, we cannot hope to define H as an infinitesmal generator of e-tH. If Y is not in L”, the only way of obtaining a result of the form II e-lHyll lip < C 1) 4 lip is to use the full hypercontractivity. Put differently, the property that allows us to obtain H as an operator on L2 is the boundedness below estimates on the H,‘s. We thus see why the field theoretic estimates of Nelson, Glimm and Federbush enter crucially in the self-adjointness proof.

PROPOSITION 2.13 (Segal). Let e- tHo be a hypercontractive semigroup. Let V E Lp for some p > 2; eevt E L1 for all t > 0. Let H, , V, be defined as in Proposition 2.12. Then for any 1 > q > (I - p-l)-‘, e-tH* converges strongly to eetH as maps from Lq -+ L@. The operators eetH are a strongly continuous semigroup on these Lq and obey:

where C and m are dependent only on H,, and q. (Note that we will eventually improve this result.)

Proof. For $ EL”, e-tH$ + e-“$ in norm, in any Lq, q < (1 - p-l)-‘. Since the e- tHn are uniformly bounded and L” is


dense in any L *, the strong convergence follows. The strong continuity of the e- 1H follows from the uniformity of the L2 convergence of e-lH$ (for # EL”) on compacts, the strong continuity of the e-lHn, the uniform boundedness of the e--fH and the density of L” in L2. The bounds follow from the bounds on the e-tHn. Q.E.D.

We are thus able to state the following:

DEFINITION (Segal). Let e-Ho1 be a hypercontractive semigroup. Let V E Lp for some p > 2, e- w EL’ for all t > 0. We define the formal sum Ho + V as the unbounded operator on L2 which is the infinitesimal generator of the strongly continuous exponentially bounded semigroup eetH defined above.

Corollary 2.8 immediately extends:

COROLLARY 2.14. Under the conditions of Proposition 2.13, -H + E(V) is a dissipative operator for some E(V) with E(V) <

--D - log I/ e-’ Ilm f or some m and D independent of V. If V is real valved, H is self-adjoint and semibounded; E(V) is a lower bound on the spectrum.

What we have discussed so far is contained implicitly or explicitly in Segal’s work [52, 531. Segal’s final and most crucial result is essential self-adjointness of the operator sum H, + V in case V is real. We give a modified proof in the spirit of our development so far. We first recall three elementary Hilbert space facts:

LEMMA 2.15. (a) Let D be dense in Hilbert space 8. Let A be self-adjoint and semibounded on Z. Then for any t > 0, e-“*D is a core for A.

(b) Let A, , A be self-adjoint semibounded operators on a Hilbert space X’ which are uniformly bounded below. If e-*nt converges strongly to e-*l for all t > 0, then Ase--Anl converges strongly to Ae-A t for any t > 0.

(c) Let A be a self-adjoint operator and suppose 4, f D(A), 4,~ q5 in norm, AC& + # weakly.6 Then $J E D(A) and # = A$.

Proof. (a) There is a straightforward proof using the spectral theorem but consider the following slick argument: if # E [e-f*D]J-, then 0 = (#, edt*+) all + E D so ectA$ = 0 (since D is dense), so

B That A&, is a weakly convergent sequence rather than a general net is crucial.


4 = 0. Thus e-tag is dense. Since it is a family of analytic vectors for A, it is a core by Nelson’s theorem [43].

(b) Since the A, are uniformly semibounded, the {e+An#} are a family of analytic functions on {t ) Re t > 0} which are uniformly bounded on compacts. By assumption, they converge pointwise on the real axis. Thus, by the Vitali convergence theorem, they converge uniformly on compacts so the convergence of the derivatives follows.

(c) Pass to a spectral representation for A. By passing to a subsequence, we can suppose +n(~) +4(x) pointwise a.e. Since A& is weakly convergent, there is a C with // A+, /I2 = J x2 1 &(x)12 dp < C. By Fatou’s lemma, lim inf x2 1 &(x)I~ = x2 1 +(x)1” is in L1, i.e., 4 E B(A). Let r) be a function of compact support in the spectral representation space. Then xrl(x)+,(x) + X$X)&X) in L1, so (7, A+,) + (7, A+) for a dense set of T’S. Since the Ij A+, /I are bounded, A+, + A+ weakly so $I = A$. Q.E.D.

Our first main result is:

THEOREM 2.16 (Segal). Let e- lHo be a hypercontractive semigroup. Suppose V is real, V E Ln for some p > 2 and e-tv E L1 for all t > 0. Then the formal sum H = HO + V is essentially self-adjoint on a subset 9 of D(H,) n D(V) and H is just the closure of the usual operator sum HO + V.

Proof. Let 9 = e-lHLm. By Lemma 2.15(a), g is a core for H. By Proposition 2.12, ~9 C Ls for all s < 03 and e-lHo# 4 e--tH$ weakly in Lp where q-l + p-l = l/2 (for any # EL”). In particular, ~9 C D(V) and Vne-tH+ + Ve-lH# weakly in L2. By Lemma 2.15(b) and Proposition 2.13, H,e@‘+ -+ He-tH$ in norm. Thus Hoe-lH+ --+ (H - V) eptH# weakly and eetH+ --+ e-IH# in norm. By Lemma 2.15(c), e-tH# E D(H,,) and H,,(e-tH$) = (H - V)(e-lH#). Q.E.D.

Remark. By using the resolvent in place of the semigroups and results we prove below one can prove the formal sum H,, + V is always the closure of the operator sum, even if V isn’t real. Also, one can prove the Lq generator of e-tH is the closure of the ordinary Lq sum if q-l + p-l > 9. Since the real V case on L2 is what arises in application, we do not discuss the details.

The next question which naturally arises is whether HO + V depends continuously on V in some sense. To prove a suitable result, we turn to proving norm convergence of semigroups (not uniformly in [0, I] however!) via DuHamel’s formula. The critical inequalities


needed are that ewtH takes L’ to L” with s > Y if t is big enough; that is the analogs of Propositions 2.7 and 2.10.

PROPOSITION 2.17. Let eetHo be a hypercontractive semigroup. Let 1 < r, q < CO. Then there is an S, an E and an m so that for all V in some L* (p > 2) with eetv E L1 all t > 0:

For any r, there is an m and a C (independent of V) so that

II e- t(Ho+V)# IIT < C”(ll epv IL2 * IT .

Proof. Use the standard weak limit argument and Propositions 2.7 and 2.10. Q.E.D.

Proposition 2.15 also carries through:

PROPOSITION 2.18. Under the hypotheses of Proposition 2.17, e-z(Ho+v) is a bounded map of Lp --+ Lp if I arg x I < (r/2)( I - / 2/p - 1 I) with bounds of exponential growth in / z / (for p, arg x $xed) at worst.

LEMMA 2.19. Let V, WE L* ( p > 2); eetv, e&W E L1, all t > 0. Let q > p. Then for any # EL”,

e-t(Ho+“)$ = e-t(Ho+W) # + I:, e-(t-~m+“)( w _ V) e-u(H,+w)~ &,

where the integral is a weak Lq integral.

Proof. Take derivatives as in Proposition 2.9. Q.E.D.

PROPOSITION 2.20. Let V, -+ V in L* some p > 2. Suppose, for eachfixed t, jl e-““a II1 is uniformly bounded in n. Let (1 - p-l)-’ < q < p. Then for some S (depending only on H, and q) e-S(Ho+VJ -+ e-s(Ho+V) in norm as operators from LQ to Lg.

Proof. Pick T so that e-T(Ho+V) is a bounded map from Lq to Lr (r-l + p-l = q-1 ) and from L” to Lq(q-l + p-l = s-l), with bounds only depending on 11 e-Sv lIm . Then, for T < t < 2T, e--l(Ho+Vn) are bounded maps of Lq to L’, LJs to Lq with bounds uniform in t and n. Let S = 2T and let # ELM.


Then

where II A IL,B is the norm of A as a map from La to LB. This proves the proposition since L” is dense in Lg.

We can now state our second main result:

THEOREM 2.21. Let e-tHo be a hypercontractive semigroup. Let V,,, + V in Lp for some p > 2, with V, real. Suppose 11 e&V, /II is uniformly bounded in n for each t Jixed. Then, for any t

in norm as maps of L2 to L 2. The convergence is uniform on compact subsets of (0, a). In particular, for any z $ spec(H,, + V),

(IT, + v,L - z)-’ ---f (H, + v - x)-l

in II 112,2 .

Proof. By Proposition 2.20, e-S(Ho+vn) -+ e--S(Ho+V) as maps of L2 to L2 for some S. By the continuity of the functional calculus, e-olS(H,+V,) + e--aS(Ho+v) for any LY. > 0 and the convergence is uniform on compacts of (0, co). This proves the statement on convergence of the semigroups.

To prove convergence of the resolvents, we use the formula, (H - .s)-l = j: @“e-Ht for any x with

// ecHt 1/2,2 < C exp[( -Re z - C) t]

and the uniform boundedness below of the H,‘s. Q.E.D.

Remarks. (1) We view this as an abstract form of Rosen’s result on the resolvent convergence of cutoff ($2”r)2 Hamiltonians [47].


(2) This result holds in a more general context: First V real is not needed and the results hold as maps of Lq to La if (1 - p-1)’ < 4 < p.

As a final result, we translate the second half of Proposition 2.17 into a statement about resolvents and derive a weak continuity requirement sufficient for use in Section 2D.

PROPOSITION 2.22. Let e-lHo be a hypercontractive semigroup. Let VEL?’ (p > 2) with e-tv EL1 (aZl t > 0). Let C and m be as in Proposition 2.17 (dependent on r). If Re z < -log(C /( e-v l]J, then

(H,+ V-,z-l:LT+LT

is a bounded map with bound only dependent on 11 e+’ jlm .

Proof. Proposition 2.17 and the formula (H - z)-’ = Jr zZte@l dt. Q.E.D.

PROPOSITION 2.23. Let e-lHo be a hypercontractive semigroup. Let V, + V in Lp (p > 2) with )I e-lvn HI uniformly bounded for each fixed t > 0. Then, for 2; with sz@ciently negative real part, (H, + V, - z)-1 -+

(HII + V - z)-’ strongly in Lq for fixed q with q < p.

Proof. Let # E Lr with r-l + p-1 = q-l. Then for z very negative,

(Ho + v, - .z)-14 - (Ho + v - .z-‘#

= (Ho + v, - z)-‘( v - V,)(H, + v - ,x)-l #.

But (H,, + V - x)-l # EL’, I] V - V, ]Iq,r --t 0 and (Ho + V, - ,z)-l are uniformly bounded as maps of Lq to La. Since L’ is dense in Lq and the ll(4, + V, - x)-l Ilq,q are bounded uniformly in n, strong convergence follows.

D. Analytic Perturbation Theory

Q.E.D.

Our last main result in this abstract setting concerns analyticity of H, + V, as an analytic function of h in some suitable sense. We recall [35], a family T(h) of closed operators on a Banach space X with z E p( T(0)) is called holomorphic at h = 0 if and only if x E p( T(X)) if / h 1 is sufficiently small and (T(h) - z))’ is a bounded holomorphic function. And we recall that for bounded operator-valued functions, norm holomorphy is equivalent to weak holomorphy [35].

THEOREM 2.24. Let D be a domain in the complex plane. Let e--1Ho be a hypercontractive semigroup and let h -+ V(A) be a map from D to Lp

580/9/z-z


( p > 2) which is analytic on D. Suppose jj e-tV(A) II1 is umyormly bounded on compacts in D. Then H,, $ V(h) (the closure of the operator sum) is a holomorphic family of operators on L2.

Proof. Let 4, # EL” and consider for z very negative,

(A - 4l)-Y~, [(f4 + VA - v - (ff, + VA” - +‘I 1cI>

= <%I + tx - z)-l& ( vy-A”v(A) ) (H” + VA,,- 2)-l Jt)* Since (H, + V,, - z)-’ + converges in L2 to (H, + V,, - z)-’ 4 (by Proposition 2.23), (V(X,) - V(A))/(h - A,) converges in Lp to - V’(h,) and since (H, + VA, - 2))’ # EL?, (7-l + p-l = &), we see that (4, (H,, -+ VA - z)-i $) is analytic at h = A, . Since the 4, # E L” are dense in L2 and the (H, + VA - z)-r are uniformly bounded for h near At,, the resolvent is weakly holomorphic and thus holomorphic. Q.E.D.

E. Extensions

Our goal in this addendum is to point out that some of our main results hold under modified conditions. In the first place, when V is bounded below (so e-VL E L1 is trivial), we only need V E L2 :

THEOREM 2.25. Let e-lHo be a hypercontractive semigroup. Let V > 0, V t L2. Then H, + V is essentially self-adjoint on D(H,,) n D(V). If V,+V in L2 and V, 3 0 all n, then (H, + V, + E)-l --j (H, + V + E)-l strongly if Re E > 0. If A 4 V(A) is an analytic L2-valued function on a domain D so that inf, (Re V(X)(x)) is uniformly bounded below on compacts, then h -+ H, + V(h) (closure of the operator sum) is an analytic family on D.

Remarks. (1) One of us (R.H.K.) has used this theorem as a starting point for an analysis of two-dimensional field theories with an :exp C+(X): 1 01 I2 < 27r interaction density [28].

(2) The basic idea is to establish that e-t(Ho+v) is a contraction on L” by using the Trotter product formula. See Ref. [28] for details.

In most field theoretic applications we have V ELM for all p (see Section 3C). It thus seems of interest to state the minimal conditions needed on H, for H,, + V to be essentially self-adjoint.

THEOREM 2.26. suppose V E Lp allp < GO, V is real and edtv E L1 for all t > 0. Let M, be a semibounded self-adjoint operator on L2 so that:

(a) For some s > 2 and some C

II edtMO IIs d ect II $ IIs ;


(b) For some r > 2 and some T > 0, e--TM0 is a bounded map of L2 to L’.

Then M,, + V is essentially self-adjoint on D(M,) n D(V).

Proof. Let us only sketch the details. By using (a), the Reisz- Thorin theorem and the argument of Proposition 2.6, we establish that the e-t(Ma+VJ are uniformly bounded as maps of LS to Lk for s > k > 2. By a DuHamel argument, the semigroups converge in norm as maps of LP to LQ for any s > p > q > 2. By (b) and Segal’s lower boundedness argument (Lemma 2.9) the e-t”ofV) are uniformly bounded as maps of L2 to L2. Our essential self-adjointness proof (Theorem 2.16) now carries through. Q.E.D.

Segal has remarked [53] that this results hold if e-iv E L1 for some fixed t = t, (and thereby for all 0 < t < to). The essential self- adjointness result also carries over without any real change in such a case

PROPOSITION 2.27. Theorem 2.16 holds if the condition eetV E L1 for all t > 0 is replaced with eetv E L1 for some t > 0.

Finally, we remark that we will prove the following abstract result in Section 4C on the relations between the La bounds of e-t(Ho+v).

PROPOSITION 2.28. Let e-tHo be a hypercontractive semigroup. Let v E I?, real, for some p > 2; let eP tV E L1 for all t > 0. Suppose E = inf gLz(HO + V) is the bottom of the L2-spectrum of H = H, + V. Then, for any q, there is a C so that:

11 e- ai,+v) / (/a,4 < CeF

for all t large. In particular, if Re z < E, then x is in the Lq-resolvant set for H,, + V for any q < co. If (H,, + V)Q = EQ for some Q E L2, then Q ELP for all q < KJ, so that inf a,,(H, + V) n R = E also.

3. HYPERCONTRACTIVE SEMIGROUPS IN FOCK SPACE

To apply the theory of Section 2 to a formal sum H,, + V arising in field theory, we need to find a representation of Fock space as L2(M, CL) in such a way that V is an unbounded multiplication operator and then we must prove four things: (1) 11 e-tHo$ II1 < 11 #Ijl for all t > 0,

(2) II e-Tffo$ II4 < C II /J II2 f or some T and C; (3) V E LP for some p > 2;


(4) e-l” E L1 for all t > 0. Since the typical field theory interactions are heuristically unbounded operators affiliated with the abelian Von Neumann algebra of the time zero smeared fields, it is natural to obtain L2(M, CL) by taking M as the spectrum of this algebra.

In Section 3A, we present a formalism for dealing with Lp(M, p) directly; this formalism will ignore the underlying points of M, which as the spectrum of a Von Neuman algebra is quite complicated. In Section 3B, we will show the quantization of any one particle operator w 3 ml (for some m > 0) is the generator of a hypercontractive semigroup. This result is not new; it is explicitly proven by Segal [43] and also follows from earlier results of Glimm [18] and a limiting argument. We provide this new proof because we feel it is more elegant to never put in cutoffs or make approximations with pure point spectra; to put it more strongly, we feel it is essential to begin to deal as directly as possible with LP(M, p). In Section 3C, we will turn to the Fock space associated with a massive Bose field in two-dimensional space-time and will present the continuous form of Nelson’s argument that interactions 6’ = Jg(x) :P($(x)): dx obey (4). We will also see that various self-adjointness statements for V are easy to prove in this formalism.

A. Coherent Vectors

To keep this paper as self-contained as possible and thereby accessible to as large and audience as possible, let us first review “abstract” Fock space in the formalism of Cook [l I] and Segal [49]. We will restrict ourselves to the Bose case. Let 2 be a real Hilbert space and let F1 be its complexification. Let 5s = (and 9% = (By g& , the symmetrited n-fold tensor product of gr . (It is the symmetrization that makes this Bose-Einstein.) Fock space is 9 = @,“=, Rfi ; thus 9 is just the symmetric tensor algebra over Fi .

Given h E 3, define the creation operator u*(h) on 9 with a*(h) : S?jj + 9-ja+1 by:

a*(h) A = l/n + 1 %+dh 0 9%). (3.1)

Here 1,5, E 27m and S, is the symmetrizer from @m Fr to Fm . It is not hard to show that any J/J, E Fm is in @[a*(h)]*) and that the adjoint of a*(h) which is denoted by a(h) (the annihilation operator) acts on #, by:

44 4, = (d+Yh, Ijrn> (3.2)


where (h, > is defined on @, F1 by

The vectors in any Sm are analytic vectors for u(h) and u*(h) and on the incompleted sum, F, of the F, (i.e., finite sums of vectors in some F,J, the a’s obey the canonical commutation relations:

b(h), a*( dl = <k g> 1. (3.3)

It is our desire to have 1 on the right side of (3.3) that led to the unantural looking 2/q in (3.2). If one defines7:

@(h) = a*(h) + a(h), (3.4a)

?-r(h) = (2i)-ya*(h) - u(h)], (3.4b)

then Q(h) and n(h) h ave all of F as analytic vectors, are symmetric and obey the Heisenberg relations:

II@V)l @Cdl = 0 =,b+% 4dlY (3Sa)

bo), @Cdl = -Kh, g> (3Sb)

on F. As a result, Nelson’s theory [43] tells us that r(h) and Q(g) are essentially self-adjoint on F and obey the Weyl relations:

ein(h)eiO(ole-in(h)e-i~(~) = &<h,o). (3.k)

If one realizes Z as L2(R, &L) so that SrCm is the totally symmetric complex-valued functions on R”, then (3.1) and (3.2) take the explicit form:

and

’ It is more usual to put l/d/2 in both CJ and v and we will use that conversion when we get down to concrete Fock space in Section 3c. To keep our formulas simple looking, we use (3.4) in Sections 3A and 3B.


Finally, one often writes Q,, for the element 1 E 9$ and calls it the Fock vacuum. This completes our review of the Cook formalism.

Following Jaffe and Glimm [20], we represent 9 as L”(Q, &) as follows: The operators {Q(g) 1 g E Z> generate a Von Neumann algebra, A, which is a maximal Abelian Von Neumann algebra with Sz,, as a cyclic vector. If we let Q be the spectrum of M, then & is algebraically isomorphic to C(Q) = L”(Q) under the Gelfand map, and 9 can be represented uniquely as L2(Q, dp) if we require fin, to go into the function 1 and that the isomorphism of& and C(Q) be spatial.s Since Q is a totally disconnected space, it might be useful (and enlightening!) to find a Hilbert space dense in Q and replace p with a (Gaussian) Hilbert space measure in one of the standard theories of Hilbert space integration [16, 24, 251. We will avoid doing this or considering points of Q directly by employing the coherent vector formalism explained below. Since p(Q) = S 1 dp = (l2, , l2,) = 1, we see that p is a probability measure.

Given a finite dimensional subspace, V, of & and a C” function of fast decrease, F, on that subspace, we define

+(F) = [j dh F(h) exp(i wa] f4 , (3.6b)

where dh is Lebesgue measure on V which we leave implicit in the symbol tJ(F). I n analogy with quantum optics [36-38-j, we call such vectors (in L2) coherent vectors. In a Hilbert space integration formalism for Lz(Q), they are the cylinder functions. If one picks an orthonormal basis h, ,..., h, in V and writes F(t, ,..., tlL) = F(Zt,hJ, then t,b(F) can also be written:

(3.6b)

where we have normalized the Fourier transform by

P(u) = j dh &-F(h)

and so its inverse is

P(h) = (27r-” j du e-+h)F(u).

On account of (3.6b), one can immediately conclude:

(3.7a)

(3.7b)

* Equivalently, the state w,(A) = <a,, , ASZ,) defines a positive linear functional

C(Q) and thus a measure on Q.


THEOREM 3.1. +(F) ELM and thus in each LP(Q). If p 3 0 (equivalently, ij F is of positive type), then 4(F) is a positive Q-space junction. If #(F) is a coherent vector, then / $(F)/ (absolute value in Q-space) is a coherent vector, explicitly:

I W)I = &h. (3.8)

Proof. Let us only prove the last statement, which we remark implies 4(F) is positive if and only if F is of positive type. ̂ ̂ A ̂Write F = F+ - F- where F, , F_ 3 0 and F+F- = 0. Then f-+(@(h,),..., @(h,)) p-(@(h),..., @(h,,)) = 0 by the functional calculus so that #(F+) #(Fe) = 0. Thus #(F) = #(F+) - #(F-) with $(F+) #(IL) = 0, #(F+), #(F-j 3 0 implies

I WI = W+> + W-) = $6, Q.E-D.

Now, let 6!! be the algebra consisting of all operators of the form J dh F(h) exp(i@(h)), w h ere we integrate over all finite-dimensional spaces and FE 9. Since J&’ is not the norm closure of a, (&I%?~ (i.e., all coherent vectors) is not dense in L” but J&’ is the weak closure of 6! so that:

THEOREM 3.2. For any 1 < p < co, the set of coherent vectors is dense in Lp. The positive coherent vectors are dense in the positive Lp junctions.

Proof. This is a general result depending only on the weak denseness of a in JH. It is sufficient to prove {afi,,} is I/ ]/,-dense in L” for each p. Let ME &? and let ii? be the corresponding L” function. By the Kaplansky density theorem, we can find A, E C!? with A, + M strongly and 11 A, I/ < 11 M 11 for ali ~1. Thus (A, - M)2n + 0 strongly for any n. In particular,

As a result, {CZSZ,,} is L2n dense in L” for any 71 integral which is enough to conclude LP denseness for any p < m. Since 1 4(F)\ is a coherent vector whenever $(F) is, the positivity statement is true. Q.E.D.

Before stating our next result, it is useful to consider the object

4.0) = 1 dh F(h) exp[i@(dh)], (3.9)


where A is a linear transformation from V to another finite-dimensional subspace, W, of 2. By a change of variable, $A(F) is a coherent vector (built from a function on W) and since being of positive type is a notion invariant under change of variable #A(F) > 0 if F is of positive type. Let us make a simple computation:

LEMMA 3.3.

(Sz, , #AW = 1 dW4 exp(-ii II Ah II”>. (3.10a)

If we consider the form h --+ 11 Ah II2 and diagonalize it so it is expressed in some basis as

and if the form h -+ 11 A-lh II2 is defined by

i

-2 aI

-*.* 1 -2 a,

and if det, A = nrEk, ai , then

(Q, , #A(F)) = (det, A)-1(27r)-“/2 j &p(u) exp(-& 11 A-4 11”). (3.10b)

Proof. Equation (3.10a) follows from the formula:

<Qo, vWYg)l Qd = exp -l/2 Ilg l12. (3.11)

To prove (3. I l), we remark that since [a*(g), a(g)] = -I[ g [I2 holds on a common domain of analytic vectors, we have

explWgN = expF~*(g)l explWf)l exp[--* II g l121. (3.12)

Using u(g) 1;2, = 0, (3.11) follows. (Alternately, (.12,, , [D(g)]” J&,) can be explicitly calculated.) Equation (3.1Ob) follows from the Plancherel theorem and elementary computations of transforms of Gaussians.

Q.E.D.


We are now able to explicitly compute Lp norms of coherent vectors:

THEOREM 3.4.

I/ ~,$,(F)l~~fl = det, A)-‘(27~)-~/~ 1 du 1 @(u)jn exp(-i /I A-lu II”). (3.13)

Proof. From (3.6b), it follows that 1 I,!J(F)IP = #(I P 1”). Thus, (3.13) follows from (3.10b) and the elementary fact llfllz = (Sz, , If 1”).

Q.E.D.

B. Quantized Operators as Generators of Hypercontractive Semi’roups

Our goal in this section is to prove a large class of natural operators in Fock space are generators of hypercontractive semigroups. As we have mentioned before, such a result is implicit in the work of Nelson [45] and Glimm [18] and explicit in the work of Segal [53]. We wish to provide a proof which by using our coherent vector formalism uses a minimal number of cutoffs.

We again begin by reviewing a general formalism of Segal [49]. Let u be a unitary on Fr . There is a natural induced map on 0, Pi by h, Q ... @ h, --f Uh, @ a-. @ Uh, which leaves the symmetric tensors invariant and thus U induces a map r(U) on 9 which is determined by r(U) rsl = U and r(U) is a unitary preserving the symmetric tensor product on 9 (i.e., I’ is the natural map from automorphisms of fir onto automorphisms of g). r induces a natural map, dr from self-adjoint operators on Fr to self-adjoint operators on 9 (i.e., on the Lie algebras of the unitaries) by

r(eitw) = exp[it dI’(w)].

We will deal below only with self-adjoints o : 2 + 8. If we pass to a spectral representation where X = L2(R, dp) and (C&(/-Z) = w(k)f(k), then H, = dr(w) has the heuristic form:

r-r, = s 44 a*(k) 44 444 where u#(k) is the operator valued distribution given by

i h(R) a+(k) d,u(k) = a#@).

A particular simple object is dr(1) = N which acts as multiplication by n on Fm : it is called “the number operator.” We also remark dr(w) Sr, = 0 for any w and dr(w) > 0 if w > 0. For such an w, one can explicitly compute exp[- t dr(w)] on coherent vectors:


THEOREM 3.5. Let w > 0 be an operator on 2 and let H, = dr(w). Then :

e -tHu[~(F)J = j dh F(b) exp[ -4(/z, (1 - e-2wt) h)] exp[@(e-Wz)] Q, . (3.14)

Proof. By (3.12), +(F) = J” dhF(h) exp[-i 11 h il”] exp[ia*(h)] Sz, . By the fact that e@‘o preserves tensor products, we see that eMtHOu* = a*(e-lwh) e- tHo on F and thus, since vectors in F are analytic vectors for a*(h), e- tHo exp[ia*(h)] = exp[iu*(e-Wz)] eetHo. This combined with e-tHoS2, = Sz,, implies

e -“““[~(F)] = j dhF(h) exp[-4 11 h I!“] exp[Lz*(e-tw)] Sz, .

Using (3.12) in reverse, we obtain (3.14). Q.E.D.

As an immediate corollary, we recover a positivity result first proven in the infinite degree of freedom case by Glimm and Jaffe [21]:

COROLLARY 3.6. Let w 3 0 and let H,, = dr(w). Then ertHo is positivity preserving on L1(Q) and thus (by Propositoin 2.1), a contraction on each Lp(Q, dp).

Proof. Since the positive coherent vectors in L1 are dense in the positive L1 functions, we need only prove ertHo[#(F)] 2 0 if #(F) 3 0. By (3.8), I/J(F) implies F is of positive type. Since w 3 0, (h, (1 - e-20t)h) is a positive form so that exp[-i, (h, (I - er2w”)h)) is of positive type. Products of functions of positive type are of positive type so (3.14) implies e-“HO[#(F)] 3 0. Q.E.D.

To prove e--lHo is hypercontractive for some class of H,, , we must look at the norm of e-IHo as a map of L2 to L4; our goal is obviously to prove that 11 edtAo EW>lll4 G c II Wh * s ince the LP norm is expressed in terms of P, it is natural to study e-tHo as it acts on fl with F lying in some fixed finite-dimensional subspace. In terms of action on F, (3.14) tells us e--tHo acts as multiplication by a Gaussian followed by a change variable. Thus, in terms of action on P’s, we have convolution with a Gaussian followed by change of variable.

Explicitly, given I’ a finite-dimensional space, let us pick orthonormal coordinates q1 ,..., qn. which diagonalize the form h --f (h, e- 2wth) and let e-2wii be the principal moments of this form. By (3.13), the L2(Q) norm of 4(F) is

(27r)-“‘2 j d”q exp( -$q2)/ P(q)12. (3.15a)


Suppose each wi > 0. Then, by (3.14),

where

e -tHo[#(F)] = j dh(KF)(h) exp[i@(e-twh)] 52,

is (Ql ,... , 4n) _ p-42 fi (1 _ ,-24)-l/2 i=l

X s

d”q’P(q’) fi exp[-I /2(1 - eKzwit)-l(qi i=l

q’i)2. (3.15b) -8

Finally, again using (3.13), the L4-norm of e-lHo[#(F)] is

(2~)~~‘~ J Pq exp(-+q’2)j a(ePwl’qI ,..,)14. (3.15c)

Equation (3.15) is equivalent to proving a bound on the map P, : L2(Rn, (2~)-~/~ e-“*&p) +L4(RvL, (27~-~/~ eddnq) given by

(ptfkl T...) qn) = (2zp’Z jd?l 4’ exP(-+d2) [fi IL&i > d)] f(%~>..., Qn’) i=l

with (3.16a)

(3.16b)

We have thus found a Fock space proof of Mehler’s formula (3.16). We remark that if one supresses the temptation of writing e-li2Q2 in the measure in (3.16a) thereby eliminating the final $qt2 in (3.16b), one sees quite clearly the structure of the kernel p, ; it is just convolution followed by a change of variable, something we have already noted is implicit in (3.14).

Since we now have essentially a finite degree of freedom problem, the general proof of boundedness fromL2 to L4 follows from arguments of Nelson [45], Glimm [18] and Segal [53], which we briefly sketch:

(1) (Nelson [45]). Let

as2(q) = (2~)-l/~ ]I, [ ps(q, $)I2 e-1/2g’2 dq’.

Let p, be the operator on L2(R) generated by the kernel p,(q, q’); then, by the Cauchy-Schwartz inequality, I( p,+)(q)1 < a,(q) 11 4 II2 , sop, is a bounded map of L2(R2) to L4(R) if (27~)-l/~ J asa ec1j2q2 dq < co. By explicit computation of the integrals [45], this is so if e2s > 3.


(2) (Glimm [18]). Following Glimm, we remark that since n kernels enter in (3.16a), we expect the nth power of any // II,,,-bound to enter; thus, to get bounds uniform in n, we need to show p, is actually a contraction. Now, we note that if the one-particle space Z were one dimensional, p, would be the kernel of e-Ns (N = number operator) so the P,~ generate a semigroup em.Vs, and, in addition, since N < I on @,“=r Fn = [J&J’. This can be used to prove p, is a contraction from L*(R, (2~)): e-lj2q2 dq) to L4(R, (2~))2 e-l/zq* dq) for 5’ > G, some universal constant; for details, see Ref. [18, p. 191. We only remark that crude numerical estimates show G < 2.

(3) (Segal [53]). Finally, we must show P, defined by (3.16a) is a contraction fromL2(R”, (2~)); e-1/2q2 dq) toL4(R”, (2~))* e-l12q2 dq). This fact, which can be proven by direct computationg, follows from a beautiful lemma of Segal which is so nice we cannot overcome the temptation presenting it to the reader:

LEMMA 3.7 (Segal [53]). Let (M, P-L) and (S, V) be two measure spaces and let T : Lp(M) --t L’(M) and K : Lp(S) + L’(S) be given by kernels T(m, m’), K(s, s’). Let T @ K be defined by [(T @ K)f](m, s) =

.f T(m, m’) K(s, s’>f( m’, s’) dp(m’) dv(s’). If T is nonnegative (as a

function on M x Ml, then II T 0 K llp,r < II T lllj,r II K l/,j,1. .

Proof. Since T is nonnegative, for any Banach space whatever Ts : Lp(M; B) + L’(M; B) given by

(T&(m) = j” TM, m’)fW) 444

is bounded by /I T Ilr,,? ; in particular, consider

T L'(s) : L"(M, LP(S)) SE L"(M x S) 4 LplI, LP(S)).

K induces a map I? : L’(M, D(S)) + L’(M, L’(S)) = L’(M x S) which only acts on the fibers, so it has bound Ij K Ijp,s. Since T @ K = I?T,pc,, , the lemma is proven. Q.E.D.

We are thus able to conclude that as long as the numbers wiT are uniformly larger than G, e--TH 0 is a contraction from L2(Q) to L4(Q). This will happen if w > cl for some c > 0; we have thus proven the major result of this section:

B J. Glimm, private communication.


THEOREM 3.8 (Glimm [18], Segal [53]). Let H,, = dr(w) with w > cl (for some c > 0). Then e- tHo is a contraction from L2(Q, dp) to L4(Q, d,u) if t 2 Gc-I. In particular, e- tHo is a hypercontractive semigroup in this case.

As a simple application of this result, we consider the relation between finite particle vectors and the Lp spaces:

THEOREM 3.9. Let + E zL . Then 4 E LP for any p < co and explicitally:

I/ *iI 2m+l < e+nn‘G II 4 112 . (3.17)

The finite particle vectors F are dense in each LP ( p < CO); in fact, for any H,, = dr(w) (w > cl f or some c > 0), the vectors in F which are analytic vectors for H, are dense in each LP (p < 00).

Proof. By the proof of Proposition 2.2, eemGN is a contraction from L2 to LZm+l which proves (3.17). To prove the last statement, fix p and note that for T sufficiently large, e-rHo is a contraction from L2 to L”, so {ebTHo# 1 # E F} is a set of vectors in F which are analytic vectors for H, . Moreover, since F is dense in L2 and e-THo has no kernel in Lq (q-l + p-l = 1) this last set is dense in LP.

Remarks. (1) If +~Gx=l~u, we still have (3.17). Since

II *iI 2m+l < II e +NmGIC, II2 d e+nmG II $ l12.

(2) By using the Stein interpolation theorem, (3.17) holds for all m. Since they are unbounded operators affiliated with J&’ we think of the fields as unbounded multiplication operators. Since (Q, , [@(h)12” Q,,) < co, we should have @(h) E LP for all p < r3. In fact, we can be pedantic and prove:

PROPOSITION 3.10. The operator @(h) (for any h E ~45’) is an unbounded multiplication operator on L2(Q); explicitly

Proof. Let P, be the spectral projection for @(h) associated with the interval [-n, n]. Since 52, E D”(@(h)) for any m,

(@(h)pnFll - (@‘(hw% in norm. Now @(h)Pn E ~2’ so @(h)P, is a multiplication operator by a function @m. By the norm convergence of (@(h)P,)92, we conclude that mn is Lzm Cauchy for

anymsoforanyp < co,mn - $j in Lp for some function @*.


Since any # E F is in all LP, $I E Om). Thus

G) 4 = lip Gn * = IiF Q(h) P,# = Q(h) 4,

since # E D(@(h)) also. Thus Q(h) = @T) on 1;. Since F is a core for @(h) and @T) is self-adjoint, O(h) = @m. Q.E.D.

Finite particle vectors are still analytic (although no longer entire) vectors for [@(h)]“; b u t f or n > 2 they are no longer even analytic for [G(h)]“. Nonetheless, we have:

THEOREM 3.11. Let A be a polynomial in @(h,),..., @(h,) (which is well defined as a multiplication operator in all LP, p < a). Then F is a core for A; in fact, if H,, = dr(w) (w > ~1, c > 0), the vectors in F which are analytic for H,, are a core for A.

Remark. It is not hard to show A < c(N + 1)” for some m, so that C”(N) C D(A) and that C”(N). is invariant under A (by the above, it is a core). C”(H,,) C D(A) will hold in general, but C”(H,) will not be invariant in general.

This last theorem follows from the elementary lemma.

LEMMA 3.12. Let (M, p) be a $nite measure space. Let V E Lp (p > 2). If .9 . d zs ense in Lq with p-l + q-l = l/2, then 9 is a core for V.

Proof. By Holder’s inequality, D(V/9) 3 Lq 3 Lw. Since La is clearly a core for V, the lemma is proven. Q.E.D.

C. Wick Polynomials in Two-Dimensional Space-Time

We now turn to specific interactions in a specific Fock space. As the one-particle space 2, we take L2(R, dk) (we think of k as a “momentum space variable” conjugate to the space part of a two- dimensional space-time). We view h --+ a(h) and (h E 2 is real-valued) h + a*(h) as operator valued distributions [64] which we write as a (k) and a*(k). We then define the operator valued distribution?

p)(x) = (277-1/2(&j-1 j e-i”5[a*(k) + u( --k)] c~(k)-~/~ dk, (3.18a)

with w(k) = (k2 + my, (3.18b)

m some fixed constant.ll

I” We form the maximal Abelian algebra JZ and the coherent vectors I/J(F) using smeared x-space fields, I = J h(x) v(x) dx.

I1 m is the mass of the “bare”, i.e., noninteracting, particle. Equation (3.18b) is just the relatavistic energy momentum relation.


Besides the normalization factors and the ~(k)-l/~ (which we discuss below) q(x) differs from @(A) in an additional crucial aspact: @ is chosen so it is self-adjoint when smeared with a real-valued K-space function, while 9 is chosen to be self-adjoint when smeared with a function real in x-space; this is the reason for the a(--K) in (3.18a).

Let w be the one-particle operator which is multiplication by w(k) and let:

H,, = dT(w) = j u*(k) u(k) w(k) dk. (3.18c)

By our discussion in Section 3B, H,, is the generator of a hypercontractive semigroup since w 3 ml.

Parenthetically, we remark that the w(k)-li2 in (3.18a) is chosen for a simple reason: If one defines

qJ(x, t) = e~y+) e--iHot = 4rr -l/2 ( ) j e-ikr+itw(k) [u*(k) + u( -k)] ,(k)-1’2 dk,

q is made Lorentz invariant by the factor OJ(K)-~/~ in the sense that F supports a unitary representation A + U(A) of the Lorentz group obeying U(A) ~(3, t) U(A)-i = ~(A(x, t)).

To get Lorentz invariant interactions, it is natural to try to work with objects like y,“(x). Unfortunately, powers of distributions are often meaningless and F”(X) is no exception. There is, however, a simple modnication of y”(x) which is well-defined. If one writes out say ~~(3, t) formally in terms of (3.18a), smears with some function of x and t tries to apply it to the vacuum, one finds that the only misbehaved term is the au* term; for T~(x, t) all a* .*a a* a .** a terms are well behaved. This suggests one define an object: F~(x, t): by ad hoc changingi any u#(k,) ... u#(kJ into a term with all u*‘s on the left and a’s on the right. Such a definition was proposed by Wick [61] and Garding and Wightman [64] have shown @(x, t): so defined for an m-dimensional space-time is an operator-valued distribution smeared over space and time. What makes two-dimensional space-time special is that for the free field of mass m [(3.18a)], the Wick product is well-defined when only smeared in space [29, 511, explicitly, one defines:

j&c) : c&v) : dx = (237-n/2 2-n’2 i. (; ) j a*@,) 0.. a*&)

x u(-kj+l) ... a( -k,) 1 (‘f k”) fi w(k&l’” dk, . i=l i=l (3.19)

I2 We adopt the symbol a# to stand for a or a* in the same way f stands for + or -.


A priori, this is only a formal definition but one has:

LEMMA 3.13. If W(k, ,..., k,) eL2(Rn), then for any j,

s W(k, )..., k,) a*@,) **. a*(&) a(-kj+J ‘-a u(-k,) dk, 0.. ak,

is well-defined applied to any finite particle vector. Its udjoint is an extension of j” W( k, , . . . , k,) a *( -k,) - *. u( k,) dk, +. * dk, deJined on finite particle vectors. If W, + W in L2(Rn) and $ E F, then

s W,(k, ,..., k,) a*(&) ... a(--kn) $!I +

s W(k, ,..., k,) u*(k,) *** a(-k,) $fl

in norm.

Proof. An elementary computation; see Ref. [4] or [20].

LEMMA 3.14. If g E L2(R), then &Cy=“=l k,) nF=“=l w(ki)F112 ELM+

Proof.

i=l ifj

and

J - dk, .*. dk, n [w(k$l]“‘+l 1 &zTk,)12 < co. Q.E.D. i+n

Notice that the fact that dki is a one-dimensional integral (i.e., that space-time is two-dimensional) is critical here and also that the L2 norm of &-w(K&‘l” is bounded by C (1 g II2 . We thus see [29] that the formal object defined by (3.19) . IS a symmetric operator on the finite particle operators. Our main goal in this section is to prove the following result ((a) has been proven by different means by Lanford and by Jaffe and Doplicher [62] and by Segal [51]; (b) and (c) are implicit in Glimm [ 181 and Segal [53]; a weak form of (d) appears in [20]):

THEOREM 3.15. Let P(X) = CzSO u,X” be a polynomial which is bounded below (i.e., N is even and uN > 0) and let g E L1 n L2(R), g 2 0. Let V be de$ned by:

V = j-g(x) : &J(X)) : dx = 5 a, Jg(x) : p”(x) : dx. 7Z=O


Then:

(a) V is essentially self-adjoint on F (and on Cm(&) n F).

(b) The closure of V, which we also call V, is a multiplication operator by a function V E LP(Q, dp) for any p < CO.

(c) For any t > 0, e-lV E L1(Q, dp).

(d) lf g has support in a set (-a, a), then eiVt is in JH(-~,~) , where A?( - a, u) is the weak closure of the bounded functions of the fields smeared with functions f with support in (-a, a).

The idea of the proof is the following: With any function h E 9, Schwarz’s space of functions of fast decrease, we associate a cutoff field &(x) = J h(x - y) $(y) dy and a cutoff interaction V, = J dxg(x) :P(&(x)): . We will see :P(#J~(x)): is a polynomial in the smeared field q&(x), so V, E LP(Q, dp) for each p (since g E Ll). We will also see that the V, are Lp-Cauchy as h(x) -+ 6(x), so there is a function v E Lp all p < co with V, -+ P in (1 lip . Then, as in Proposition 3.10, V is essentially self-adjoint and its closure is P. Finally, we will prove (c) following Nelson’s cutoff argument [45] by showing each V, is bounded below and using explicit estimates on

II v - Vh lip .

LEMMA 3.16. Let h, be any sequence of functions in L2(R) so that (i) j/ h, Ill0 is bounded, (ii) h,(k) -+ 1 pointwise a.e. Let P(x) be any polynomial and let g E L2(R) n Ll(R). Let

v = s

g(x) : P(f$(x)) : ax

and

V, = .r

g(x) : P(&,(x)) : dx.

Then:

(a) If Z/ is any finite particle vector, V,# -+ V# in I/ 11.

(b) The V, are multiplication operators in LP and are Cauchy in each Lp(Q) space and in f&t

II V, - V, II 2”+l < ekdG II V, - V, 112, (3.20)

where d is the degree of P.

Proof. The effect of changing s :4”(x): g(x) dx into J :&“(x): g(x) dx in the definition (3.19) is to replace the kernel &x:%1 ki) l-I:=, w(ki)-1/2


with && Ki) nFzk, w(K~)-~/~ &kg). Under the assumptions on the &‘s, the kernels converge in L2(R*) so (a) follows from Lemma 3.15. We will see below [Eq. (3.21)] that X&“(X): is a polynomial in b(x) so it is a multiplication operator in each Lp. Since g E L1, V,, E Lp. We know V,Q, -+ VQ,, in 11 11, so V,sZ, is jj JI-Cauchy, i.e., V,(Q) is L2(Q)-Cauchy. Moreover, from (3.19), VJ& E @L1 9i so (3.20) follows from (3.17). That /) V, - V,, lip --+ 0 follows now from the L2(Q). Cauchy condition. Q.E.D.

COROLLARY 3.17. Let P, g, and V be as in Lemma 3.16. Then (a), (b), and(d) of Theorem 3.15 follow.

Proof. The V, converge in all Lp(Q) norm to some function v (p < co). By (a) of L emma 3.16, r# = lim V,# = V# for any $ E F and by Theorem 3.9, P is essentially self-adjoint on F. Finally, to conclude (d) of Theorem 3.15, pick h, so that h,(x - y) has support in (-a, a) whenever y E sup g. Then eiV*l E &(-a, a). Since V, -+ V strongly on a common core, eiysl + eiVt strongly by the Trotter-Kato theorem. Thus eir’l E JZ-n,a) . Q.E.D.

Finally, we need to prove epvt E Ll(Q) and this requires explicit lower bounds on the Vn’s which in turn require explicit formulas for : Tag: in terms of vhfi(x). An expression for I .** &x,) in terms of :$a@ *.. vi,(xi,): is due to Wick [61] (and is known universally as Wick’s theorem). One can invert Wick’s theorem to express the Wick products in terms of ordinary products. This has been done explicitly by Cainiello [7]. (S ee also Eg. (3.24) of Ref. [64]). If one then smears with h and looks for :vhm(x):, one finds

rmiz1 :y)&m(x): = c (-1)r m! 2-’ (m _ 2rl! r! II ~(4 Go mnwi+~~. (3.21)

r=o

Equation (3.21) and elementary algebra, then implies:

PROPOSITION 3.18. Let P(X) b e a polynomial of degree d which is bounded below. Then, there are constants c1 and c2 so that

:wd~)): 2 -cl - C211d4Q,lId. (3.22)

In particular, if hK is the function whose Fourier transform is the characteristic function for {h 1 / k I < K} and V,,, = j-g(x) :P(~+(x)): dx for g E L1 n L2, g > 0, then for K big (independent of g):

v,,, 2 II g M--cdl% w21*


Proof. Equation (3.22) is arithmetic. To prove (3.23), we compute

II dhc) Q, /I2 = (477)-l II @P(&4k)-1’2)l12

= (477-l j,,, >K w(k)-1 dk < C(log K). Q.E.D.

To apply Nelson’s method we also need to know (1 V - V, 11 21n2:

LEMMA 3.19. Under the assumptions of Proposition 3.18,

for some a > 0 and C, independent of g.

Proof. By (3.20), we only need prove the formula when m = 0. Since V is a linear combination of objects of the form Jg(x) @(x): dx, we need only consider that case, By (3.19),

By taking 01 small, this last integral is finite and bounded by C” 11 g 11s (as in the proof of Lemma 3.14).

Remark. Rosen [47] has shown that (3.24) with K? replaced K-l(log K)d is true. All one needs is some 01 > 0.

We can now prove the result implicit in Nelson [45]:

THEOREM 3.20. Let P, g be as in Theorem 3.15 and let

V, = g(x) :P(+(x)): dx. s

Then for any t > 0, and any q < CO, e@’ E Lq(Q, dp). Moreover (forfixedp), II e+’ II9 h as a bound only dependent on t, q, 11 g Ill, and 11 g 112.

Proof (Nelson [45]). W e only sketch the proof; for details, see Ref. [45] or [18]. The idea is simple; for any X > 0, pick K so that x = II g lll[--c,w W”‘“l.


Then, by (3.23),

For fixed K, one picks m to minimize the bound on /I V - V, II& given by (3.24). One then has sufficient control on ~{p j V(p) > -X - l} to prove Se-vfp dp < CO. Q.E.D.

4. SPATIALLY CUTOFF TWO-DIMENSIONAL BOSE FIELD THEORIES

In this section, we wish to discuss in detail some aspects of the applications to :P(F): field theories in two-dimensional space-time. Many of the results we describe have been obtained for (v”)s theories by Glimm and Jaffe [20-221. We describe our results explicitly for two reasons. First, we wish to emphasize which results depend merely on discussing H, + Y with e--tHo hypercontractive and V E LP (some p > 2), e-l” eL1 (all t > 0), which results depend on the additional property that H, = dr(w) with w > c1 (some c > 0) and which results seem to make essential use of higher order estimates and which are thereby not accessable by our methods. We also wish to begin a study of coupling constant analyticity.

A. Lower Bounds and Self-Aa!jointness

By combining Corollary 2.14 with Theorems 3.8 and 3.15, we immediately find:

THEOREM 4.1. Let q(x) b e a two-dimensional free field of mass m. Let V = Jg(x) :P(q(x)): d x,whereg~LlnL~;g>OandP(X)isa polynomial which is bounded below. Let N, = dr(wT), where w is the operator of multiplication by w(k) [(3.18b)]. Then N, + Y is bounded below for any T > 0.

Remark. Such N, + Y estimates have already been noted by several authors, e.g., Jaffe [31].

In terms of the Glimm constant G one can explicitally write (for h > 0):

I!?,, G inf[spec(HO + hV)] > - & log(L$ , e-zGvn/ms2,). (4.1)

This should be compared with the formal perturbation theory result

161: 4, = --log (Q, , T [exp - I", v(t)] Go). (4.2)


We note the formal similarity and remark that (4.1) has a formal perturbation expansion in terms of connected diagrams. Since

<Q, 9 eaYQ,)(a > 0) is not finite, this formal perturbation series diverges; this fact illuminates the known fact that the formal perturbation series coming from (4.2) diverges [30].

Our essential self-adjointness result (Theorem 2.16) implies:

THEOREM 4.2. Let V be as in Theorem 4.1 and let HO = dr(w). Then HO + V is essentially self-adjoint on D(H,,) f~ D(V). Moreover, the following sets are cores for H13:

(a) @(H) n C”(V);

(b) a(H) n Cp<mLp(Q, dp); (c) P(N) n Cm(V) n D(H);

(4 F n W%);

(4 WY n WG-

Note. So far as we can determine, one cannot show D(H02) n D(H) is a core for H without proving some higher order estimate. It is likely that without some smoothness assumptions on V (i.e., in the abstract setting of Section 2), one cannot prove such a result14. In the (F~)~ case Glimm-Jaffe [20] and in the (‘p2n)2 case Rosen [48] have proven Cm(HO) is a core for H,, + H1( g), by utilization of higher order estimates.

We remark to the reader that essential self-adjointness enters crucially into one removal of the spatial cutoff as follows: One expects influence in the theory to propagate only at the speed of light, so Guenin [26] suggested [exp i(H, + HI( g))t] A[exp - i(H, + HI( g))t] shouldn’t depend on what g is outside the interval15 [a - / t 1, b + 1 t 1, if A is in the Von Neumann algebra @(a,b) generated by the spectral projections of the fields and conjugate momentum smeared with functions having support in (a, b). Segal [50] then showed that Guenin’s suggestion could be proven by using the Trotter product formula for exp i(H, + H,( g))t and the fact that exp iH,( g)t E JzY(~,~) if g has support in (a, b). Essential self-adjointness is then crucial since it is needed to justify the Trotter product formula. We do not present the details of the Guenin-Segal proof since the discussion in Jaffe-Glimm [20] cannot be improved upon; we commend it to the

I3 We use b%!(H) to denote the analytic vectors for H. I4 However, without any smoothness assumption on V, Gl(Ho) A Cm(V) is dense if

VisinallL”,p < co. I6 We have set c, the speed of light, equal to 1.


reader interested in the details. We however note the final result in a slightly stronger form suggested to us by J. Glimm16:

THEOREM 4.3. Let g,(x) be positive, piecewise continuous functions in L1 n L2 so that g,(x) -+ h uniformly on compacts. Let:

aA4 = [exp W& + ffdgNl4exp - W4 + f4(d).l

with

Wg) = Jg(4 :%4x>): dx,

where P is a polynomial which is bounded below. Then,

exists for any A in ??J, the norm closure of u %(a,b) and is independent of the sequence g, chosen.

Thus, one can use g%(x) = X exp(-x2/n) for example and so let g -+ X by a family with an analytic interpolation. By our analytic perturbation theory (Section 2D) and the analytic perturbation theory for eigenvalues [35], one obtains a family of approximate vacuum states w, (see Section 4B) analytic in a neighborhood of the real axis, with the “thermodynamic limit” obtained by K --+ 00.

As a final subject in this section, we see what we can say about domains for the fields. Given a function f in L2(R), we define

q~(f, t) = eix%p(f) eciHt. (4.3)

Let E(H) be the set of entire vectors for H. For the sharp time fields (4.3), one has:

THEOREM 4.4. Let H = HO + V as in Theorem 4.1. Then, for any f E L2, any$xed t and any n, E(H) C D(p( f, t)“) and is a core for F(f, t)“.

Proof. s ince eiH” leaves E(H) invariant, we need only prove the result for t = 0. Since p?( f )” is an operator in LP, all p < co, we need only prove E(H) is dense in some Lp, p > 2. But E(H) is dense in L2 and e-l”[c(H)] = E(H). Since e-l* takes L2 into L2+r, we see E(H) is dense in L2fs. Q.E.D.

IS Private communication.


Remark. We could just as well deal with vectors of compact energy as with the larger set E(N).

In space-times of dimensions bigger than 3, there are strong indi- cations that the sharp time smeared fields won’t be well-defined but that fields smeared in space and time will be well-defined. It is thus of interest to consider such objects even in the two-dimensional case. Given f E Y(R2), we define

df) = j- dt dft 9 t),

where ft is the function x +f(x, t) in Y’(R).

(4.4)

It is not hard to see that y(f) is defined on E(H) but a proof of essential self-adjointness requires bounds on the fields discussed in Refs. [21] and [48].

B. The Vacuum Vector for the Spatially Cutofl Theory

Our first goal in this section will be to prove that a class of operators H, + V have ground states by a method which is essentially an abstraction of the method used by Jaffe and Glimm in the (y4)2 case [21]. We then present a simplified proof of the nondegeneracy of this ground state whose basic principle is again due to Jaffe and Glimm [21], but which uses crucially the irreducibility isolated by Segal [52]. We finally discuss, somewhat speculatively certain properties of this vacuum vector” and of the limiting state. Our basic existence theorem is:

THEOREM 4.5. Let L2(Q, drJ) be the Q-space associated with some Fock space. Let w be a one-particle operator with w > cl and let Ho = dI(w). S pp u ose V is a multiplication operator with V ELP some p > 2 and eeV1 E L1, all t > 0. Then II, + V has discrete spectrum in (E, E + c), where E = inf o(II,, + V). In particular, II, + V has a ground state (eigenvector of eigenvalue E).

Remark. This result depends critically on the combinatorial structure of Fock space and is surely not a result true in some abstract hypercontractive setting.

The idea is simple. One approximates H, + V with operators, %,n + v?L and uses the theory of Section 2C to prove (H,,, + V, + X)-r + (Ha + V + A)-l in norm if X is large. Each H,,, + V, has discrete spectrum in [E, , E, + c) and so the theorem follows from:

I’ Segal [53] has announced the preparation of a paper on the existence and uniqueness of the vacuum vector.


LEMMA 4.6. If (A,L + A)-l + (A + X)-l in norm and A, has discrete spectrum in [E, , E, + c) with E, = inf o(A,), then A has discrete spectrum in [E, E + c).

Proof. For the details, see Lemma 5.4 of Ref. [23]. The idea is simple: since

4, -j E, (A, - E, + /I)-’ + (A - E + h)-l

for h > 0, so we can suppose E, = E = 0. For an operator B > 0 to have discrete spectrum in [0, c), it is necessary and sufficient that f(B) be compact for any function f which is continuous with support in (-co, c). By continuity of the functional calculus, f (A, - E,) -+ f (A - E) in norm so discreteness of A, - E, in [0, c) implies discreteness of A - E in [0, c). Q.E.D.

Given w as in the theorem, realize the one particle space p1 as L2(R, p) so that (wf)(k) = w(K)f(K) with TV a finite Bore1 measure. For each n, we define a countable Bore1 partition, 9% , of R inductively obeying:

G) g?L+1 is a refinement of Y)n ;

(ii) For any set A E 8, , max(o r A) - min(w r A) < 2-“;

(iii) Each A E 9% is a subset of (m2-%, (m + 1) 2-“1 for some m = -292, -2%. + I,..., n2n or it is a subset of (- 00, -n] or (n, 00).

(iv) For any K, and any n, there are only finitely many sets A E 9, with min(w r A) < K.

It is not hard to construct such partitions. (iii) is demanded so that Proposition 4.7 (a) below will hold, ( ii so that Proposition 4.7 (b) will ) hold and (iv) so that the approximate Hamiltonians H,,, will have compact resolvents on certain invariant subspaces.

Let A?,, be the set of functions which is constant on each A E @, and let P, be the projection onto Z* . Finally, define H,,,, = dI’(w,), where W, is the function Pnw, i.e., the function whose constant value on A E 8, is p(A)-1 JA w(K) dp(k). We remark that We > 6.

PROPOSITION 4.7. (a) P, -+ 1 strongly in Sl ;

(b) Il(N + 1))““(Ho - H,,,)(N + I)-1/2 /I + 0 as n --+ co.

Proof. (a) We need only prove u ran P, is dense since {P,} is a monotone increasing family of projections. By condition (iii), the characteristic function of any rational dyadic interval is in some ran P, so the union is dense.


(b) By condition (ii), /I w, - w 11 < 2-n so -2-“(N + 1) < U(W~) - dr(w) < 2-“(N + 1) which implies (b). Q.E.D.

We can now apply the theory of Section 2C:

PROPOSITION 4.8. (a) Suppose WE Lp(Q, dp) for some p > 2 is real with e-lW E L1 for all t > 0. Then for h suficiently negative \l(H, + W - A)-l - (H,,, + W - A)-l I/ + 0. If Z3 is a set of LP functions, with (11 W lip 1 WE B} and (11 e-lW Ill I WE B} are bounded, the convergence is uniform in Wfor WE g.

(b) Suppose W,+ WinLp(Q,dp)forsomep > 2with{I e-lwnlll> bounded for each f. Then for h su.ciently negative,

ll(yl + w - h)-l - (f&L, + w, - A)-l /I - 0.

Proof. (a) We must only show

ll(ff, + w - w&l - f-qJ,n)(f&, + w - X)-l I/ - 0.

Since the e&Ho., are hypercontractive with constants independent of n, the Ho,, + W are uniformly bounded below, so we need only show Il(H,, + W - X)-1/2(H,, - H,,,)(H,,, + W - h)-l12 11 --f 0. By Propo- sition 4.11, this follows if we can show Il(N + l)‘/“(HO,, + W - h)-li2 11 is uniformly bounded in n or equivalently that

(f&n + w - A) 3 C(N + 1) (4.5)

for some C independent on n. But for C sufficiently small H,,, - C(N + 1) are a family of generators of hypercontractive semigroups with uniform constants. Thus, an inequality of form (4.5) holds, so the first part of (a) is proven, Clearly, all estimates only depend on II W/I, and Ij e-tw Ill so the rest of (a) follows.

(b) BY (4, lI(K, + W, - 4-l - (Ho,, + W, - 4-l II --f 0 as n -+ co uniformly in m and by Theorem 2.21,

II(& + w - X)-l - (H, + w, - X)-l /I - 0

so the result is proven. Q.E.D.

All that remains to prove Theorem 4.9 is to show we can find I’, + I’ in some LP (p > 2) with // e-o’n l/r bounded in such a way that H,,, + V, has discrete spectrum in [E, , E, + c).

Let F(m) be the Fock space built on yt” , i.e.,


and let 9(n,l) be the Fock space built on Sal. Since Z% @ Xm’ = 9r , we have

9 = &F(n) @ FW),

9tn) and 9(n,l) as Fock space have Q-space realizations,

Fn’ = L2(Qcn), dtLn) and g(%‘- ) = Lz(Q’(n), dv,).

Thus since

and

L2(Q, dLL) = L2(Q’n’, d& @ L2(Q’cn’, dv,) = L2(Q!‘“’ x Q’fn), dpn @ dv,),

we have Q = Qtn) x Q’cn) and dp = dp, @ dv, . Given W E LP(Q, dp), define 7~~ WE L*(Q, dp) by:

(rrnW)(q(@, q’cn)) = J’ dv,(q”‘) W(@), q’cn)) (4.6)

(i.e., VT, W only depends on q, not q’; nn W is obtained by averaging W in the Q’-direction).

PROPOSITION 4.9. (a) z-a is a contraction on each Lp.

(b) If e-” E L1, then ervnY E L1 and

II e-‘-” Ill < II e-” Ill .

(c) If V E Lp for p -C co, then ST-~ V -+ V in Lp.

Proof. (a) 7ra is clearly positivity preserving, self-adjoint on L2 with ~~1 = 1. Thus, (a) follows from Proposition 2.1.

(b) Since the geometric mean of a quantity is smaller than the arithmetic mean:

we have

(c) Since the z=~ are all contractions, we need only prove TT,V --f V on a dense set of Vs. Since rTT,r, = YT~~~,, = rrn for m > n


(by (i) of the properties of P). 7r,V -+ V if V E rad 7~, for any n. Let A’, be the algebra of bounded functions of smeared fields smeared with functions real and in Y& . Since P, -+ 1, v JZn is weakly dense in A! and thus Lp dense in Lp(Q) (p < co) as in Theorem 3.2. Since FV E &In implies n,W = W, U A!, C U ran T,(LP) so r,V -+ V on a dense subset. Q.E.D.

The proof of Theorem 4.5 is thus reduced to showing H,,, + V, has compact spectrum in [E, , E, + C) with V, = v,V. Let us define 19(n) by

pn,i’ = c 0’ $inn)*

We can identify 9tn) with a subspace of F by

g-(n) s F(n) (g c c $pl) @ $mt,u*

Under this identification, [%(n)]l = $Jn) @ J-F(~). We would like to consider the action of Ho,, and V, under this breakup:

PROPOSITION 4.10. (a) H,,, leaves Fn) and Wn)l invariant and

Ho,, r F(n)1 = Ho,, r (S--(fi) 0’ S(n))

= (H,,, p 5-q 0 i + 1 0 (Ho,, rl P(n)).

(b) V, leaves 2Wn) and Fn)l invariant and

V, r (S(nJL) = v, p (F(n) @ s(n)) = (v, r s(n)) 0 1.

(c) H, = Ho., + V, leaves Ftn) and Fn)l invariant and

H, r F(n)1 3 E, + C,

where E, = inf spec(H,).

Proof. (a) Since w, leaves G’& and Zm’ invariant, Ho 12 = dT(w,) leaves Fn) and .FnJ) invariant. By self-adjointness, & ‘hwL is left invariant. Since I’ preserves 0, dT(w) acts on Fnf @ FnJ) as H,,, @ 1 + 1 @ Ho,, (by Lebnitz’ rule).

(b) Since V, is independent of q’cn) and since f @g in F(n) @ s(n,l) is (f 0 g)bh 4’) = fk) g(d), we have

Ilvnu 0 m79 4’) = ~n(df(d‘fh’)

= Wnf 0 81(49 4’).


(c) H, r PcTLJ1 = (H, r ~9~)) @ 1 + 1 @ (H,,, r kern)) by(a) and (b). We have H, P Fcnf > E, and HO,, P -LF(n) >, c since 6~~ 3 cl and J-Fn) is the orthogonal complement of C in flcnyl). Thus Hnrsn 3c+E,. Q.E.D.

We now complete the proof of Theorem 4.9 by proving H, P 2Fn) has compact resolvant:

PROPOSITION 4.11. (a) HO,. r Fn) has compact redvent;

(b) H, r Fn) has compact resolvent.

Proof. (a) If A is a one-particle operator with eigenvalues with A, + co and each A, > c > 0, then U(A) has eigen-

kks:“~, (A,)& ; (A + h >” *a* i.e., has discrete spectrum. Since w, r Y& has compact r”e&?yFit (by (iv) of the properties of P), HO,, r 9(n) has compact resolvent.

(b) The Weyl min-max principle says that if A is bounded below and self-adjoint, then

has the property that either pn(A) is the n-th eigenvalue from the bottom of the spectrum of A or it is the bottom of the essential spectrum and in the latter case, &A) = F,+~(A) = -0.. It follows that A has compact resolvent if and only if pm(A) -+ CO as n -+ co. Write I’ = V+ - I’- with V+ , VP > 0. Since 1/2(H0 - VP) is bounded below, I($, JbJ>l G 1 PC+, H,&) + b(#, Q> so:

Thus,

so pn(Ho,n + vn r S(n)) -+ co by (a). This proves H, r Sn) has compact resolvant. Q.E.D.

We have thereby completed the proof of Theorem 4.9. We would emphasize once more that this proof is essentially an abstraction of that used by Jaffe and Glimm in the (P’“)~ case.

We next turn to the question of uniqueness of the vacuuml’; our proof depends crucially on the irreducibility isolated by Segal [52]


and a comment by him in Ref. [52]; it is basically merely a technical improvement of the Glimm-Jaffe proof of vacuum uniqueness [21], a proof whose basis is the classical theorems of Frobenius [17], Jentsch [34] and P erron [46]. We first prove a general result of the Frobenius type.

THEOREM 4.12. Let (M, CL) be a measure space. Let H be a semibounded self-adjoint operator on Lz(M, p); H 3 c with two properties:

e-tH $.‘(-J

eeiH is positivity preserving, i.e., if f > 0 (pointwise), then

(j) (ectH} t,,, u L”(M) form an irreducible set.

Then dim{$ / H# = ct,!r} < 1 and if HI,J~ = c1,4, then $J is a cyclic vector for L”(M).

Proof .18 We first note that if H# = c+, then H 1 # j = c 1 #J /

(where I # I (4 = I IcI(x>I), for

(I 1cI I>, e-tH I 4 I> 3 (~4 e-““$> (by (a))

= e-tc = /I e-tH II,

We next show if Hz,b = CZ,!I, then #(x) = 0, a.e. or #(x) = 0 a.e., for let S = {f j f (x) #(x) = 0 a.e.}. We must show S = {O> or S = L2(M), so by (b) it is enough to show S is left invariant by e-fH and L”(M) since S is a closed subspace. S is clearly left invariant by L”(M). Let S, = {f 6 S If >, O}. Since S = S, - S, , it is enough to show S, is left invariant by e--fH. Let f E S, . Then

(e-““f, I * I> = (f, emtH I * I> = e-Yf, I * I> = 0.

Since e-‘“f and j + / are positive, we conclude e-“f E S, . Thus, we conclude # # 0 a.e. or $ = 0 a.e.

Finally, we note that since / # / is an eigenvector, so is 1 $ I - #. Thus\+/ -#is0 a.e. or = 0 a.e., i.e., * is positive a.e. or negative a.e. It thus follows that two orthogonal eigenvectors are impossible and that $ is cyclic for L”(M). Q.E.D.

To apply this to the field theory case, we need the following:

THEOREM 4.13. Let H,, = dr(w); o > J. Let V E Lp(Q), some

I8 One can use OUT techniques to prove e- tH is strictly ergodic and SD fall back on the proof of J.&e-Glimm [21].


p > 0; e-V1 E Ll, all t > 0. Let H = H, + V, the closure of the operator sum. Then:

(a) (Jaffe-Glimm [21]) e-fH is positivity preserving on L2(Q, dp);

(b) (Segal [52]) J&’ u {erfHjlg, is an irreducible famiZy.

Proof. (a) By Theorem 2.21 and the Trotter-Kato theorem, if V, are bounded Lp approximations of V,

exp(--tH) = s;@ exp[-t(HO + V,)].

By the Trotter product formula:

--tH e = s-lim lim (e~tHO~nLe-tV*~m)m~

n-tm tn+m

Since ePHo and ePvn are all positivity preserving, so is e&H.

(b) As in (a), applying Theorem 2.21 to W, = V - V, ,

e--tHo - _ s-lim lim (e-tHime+tV~L~n)m, 1L’a2 m+m

so [A u (e-lH)]” 3 [A u (e&Ho)]“. Let A E [A u (e-tHo)]‘. Then A E A’ so A E &%’ since & is maximal Abelian. Since A commutes with e--tHo, we see e-lHo(AQO) = An,, , so /IQ,, is an eigenfunction of e-@‘o with eigenvalue 1, i.e., A&?,, = c&‘,, which implies A = cl. Thus [A! U (e-lH)]” 3 (cl) ’ is all bounded operators. Q.E.D.

COROLLARY 4.14. The ground state in Theorem 4.5 is unique and is (a.e.) strictly positive Q-space function.

Let us denote this “vacuum” state as fiH. We next turn to some of its simple Q-space properties:

THEOREM 4.15. Let H, be the generator of a hypercontractive semigroup and let V E LP (some p > 2); e-“l E L1 all t > 0. Let C$ be an entire vector for H = H, + V. Then 4 E Lq, all q < 00; in particular, any eigenfunction of H (e.g., 52, in the context of Theorem 4.5) is in every Lq, q < 03.

Proof. For any t, I$ = e-tH$t for some #t E L2. Since rad e-tH C Lq, if t is big enough, the result follows. Q.E.D.

One immediate consequence of the LP properties of the vacuum is:

COROLLARY 4.16. Let H, be the generator of a hypercontractive


semigroup and let V E LP, some p > 2; e-“” E L1 all t > 0. Let 4 be an entire vector for H = H,, + V. Then 4 E D(HJ. In particular,

QH E wf4J-

Proof. + = e--tH$ with I,!J E LP, all p < co. Thus, by our argument in Theorem 2.16,+ E D(H,).

Remarks. (1) If VELP, all p < co, then 4 need only be an analytic vector for H in order that Corollary 4.16 hold.

(2) For (@4)z , perturbation theory suggests 4 $ D(H3/2) [20], so we can’t much improve on 4 E D(H,).

(3) In the (65”)s case, this result has been proven by Glimm-Jaffe [20] and in (@z”)z by Rosen [48].

(4) On the surface, one would think of GH E D(H,) as following only from some sort of higher order estimate (as used by Glimm-Jaffe and Rosen), but, in fact, we have used no higher order estimates.

One can also make statements about LP convergence of the vacuums of H, , thereby extending a result of Glimm and Jaffe [21], who prove L2 convergence:

COROLLARY 4.17. (a) Let V, + V in Lp (some p > 2) with ]I e-“nt II1 bounded f or each t uniformly in n. Let Ho = dP(w), w 3 cl. Let H, = H,, + V, ; H = H,, + V. Then sZnn -+ Q, in each L*

(Q < a).

(b) If Ho,, = dP(w,) with each We > cl and with

II (N + 1p2 (4 - f&J.,) (N + IF2 II - 0

asn+ co andzfH, = H,,, each L* (q < co).

+ V, , H = H,, + V, then 52, --t Q, in

Proof. In either case, (H, - E)-l --f (H - E)-l in norm, so by the phase condition (Sz,, , a,) > 0, we have Sz, ---t L?, in L2. By arguments in Section 2C, eeHnT --t e--HT converge ?n norm as maps from L2 to L* if T is large enough. Thus, e-EnTQHfi -+ eeETS, in L*. Since E, + E, the proof is complete. Q.E.D.

Before leaving the subject of the LP properties of the vacuums, we would like to point out that LP estimates might be of use in studying the field theory without cutoffs. Explicitly, let

H(g) = Ho + s g(4 :Wd4): dx


and let Q, be the vacuum of H(g). Glimm and J&e [22] have proven there is a sequence of cutoffs, g, -+ I, so that the states

%&(A) = wg, Y AL’& on % converge to a state w in the W*-topology. Moreover, the representation, r, of % generated by w via the G.N.S. construction is also locally Fock, i.e., r r @(a,a) is unitarily equivalent to the Fock space representation restricted to %ta,a) [22]. In particular, by using this unitary operator, one can define smeared fields 4,(h)

on %hYS J the underlying Hilbert space for V. J&e and Glimm also prove that the G.N.S. vacuum Qphys E D{+,,(h)). For reasons related to the Wightman axioms [58], one would want QPhys E D(+,(hi) ... &(?r(hk)) for any set h, ,..., h,< of functions on space-time. One is also interested in the “equal” time fields, i.e., SZphJrs E D(&,,(h,) a** 4Jhk)), where the hi are functions of space alone [see (4.3) and (4.4)].

We should like to make a few speculative remarks about the relation of D-estimates to this domain question. We will first obtain a result that follows from a uniform estimate on /I Qg, lip (p > 2). We feel such an estimate is almost surely false as we shall explain, but we will conjecture that a weaker estimate holds with esssentially the same consequences. For simplicity, we first state the consequences of the stronger but probably false estimate:

PROPOSITION 4.18. Suppose 11 Szg, I& (in D’(Q, dp)) is uniformly bounded as n -+ CO for some p > 2. Then L?phya E D(&,(h,) .** +,(h,)) [equal time fields] for any set of functions hi with

.r dk @#z)l” w(k)-1 < co.

In fact:

Proof. Let Pk: be the spectral projection for 4(h,) .*a +(h,) onto the interval (--K, A). It is enough to prove w(P&(h,)* *+a +(h,)*]

W4 . -. d&J> is bounded independently of k. Since w, + w on bounded operators in A, we need only show w,(Pk[+(h,)* *me +(h,)]) is bounded uniformly in TZ and k. This follows in turn if we can

show ~,W*(hm) --a 9bn>l) is bounded uniformly in n. But, since

A = +*(h,) a.. $(h,) ED(Q)&l + 2p-1 = I),

we have I w,(A)I G II A /IQ II -% II; which provides the needed bound. The convergence of the vacuum expectation values is not very hard.


Remarks. While we cannot use this method to say anything about the unequal real time vacuum expectation values, we can say something about the “imaginary” time vacuum expectation valueslg, w(+(h,) eiH(tl+icl) .*. eiH(tn+ir+(h,+l)) with l 1 ,..., em > 0, if we can also obtain uniform estimates on e--t(H’+-Ec) as maps of L2 to L2+s.

We suspect in fact that as g---t 1 the // Sz, \lp are not bounded if p > Zzo. For as g + 1, more and more modes are “coupled essentially to Q,“; thus, we expect 11 Q? lip to grow as Cn(@ where C > 1 and H(g) is the size of the set on which g is 1. There is in x-space a coupling of the modes associated with disjoint regions of space but it is well-known to be a coupling which falls off exponentially with the distance between regions. This suggests the following picture: Fix a, b and consider the set S of one-particle functions h E 3 whose x-space representation has support in (a, b). As in Proposition 4.9, we have

LYQ, &I = LYQs x Qs', 4, 0 W,

where

As in Lemma 4.9, define rrS : LP(Q, dp) -+ LP(Q, dp) by averaging in the QS’ direction. As n increases, we expect the additional wiggles in sZgn should concentrate in the Q’ direction because of the falloff in the coupling of the QS modes to the new modes introduced by increasing n. Thus we conjecture that ~~(522) will have bounded Lp/2(QS, dpS) norms. This would imply Qphys E D($,(h,) *-* $,(h,)) as long as h 1 ,*.*> h, all have compact support.

As support for this picture which we conjecture we remark the following: Because of Haag’s theorem Gn2 cannot converge in Ll(Q), but by the local Fock property proven by Glimm-Jaffe [22] n,(S2,2) does converge in Ll(Q); some smoothing due to QS’ averaging is already in evidence-we see no reason for there not to be enough smoothing for the x,(G~~) to converge in all Lp (p < ‘x)).

C. Coupling Constant Analyticity

As a final subject in our general study of spatially cutoff (~2”)~ theories, we should like to examine the analyticity in K of the various objects associated with Ho + KV. After showing full cut plane

ID These are the analogs of the continuation of the Wightman functions into the forward tube.

*O Thus, the VanHove phenomenon, P, + 0 weakly in La [60,32] which is not yet proven in (@p”)z , does not hold in LP. Since Q, is L2-normalized, this is not surprising.

5w9/2-4


analyticity of the resolvent we study the ground state energy and vacuum vector in the right half-plane. In particular, we will prove the asymptotic nature of various perturbation series and we will make one comment on the convergence of the Pad& approximants to the ground state.

Our general analytic perturbation theory (Section 2D) implies H, + KV is an analytic family for Re K > 0. But Jaffe’s study of the finite number of degree of freedom casezl suggests one try to prove analyticity in the cut plane, K # negative real. In fact, this can be proven and, in fact, could have been proven after Glimm’s lower boundedness paper [18], for:

THEOREM 4.19. Let H, and V be self-adjoint operators on some Hilbert space Z? with H, 3 0. Suppose V = V,- - V- where

(a) V+ , V- 3 0 (b) VP is a tiny form perturbation of H, , i.e., Q( V-) 1 Q(HJ

and for all a > 0, there is a, b so that

for all # E Q(H,). D fi e ne the quadratic form t(K) = H, + KV on

Q(Ho) n Q(V+) f or K in the cut plane K # negative real. Then t(K) is a holomorphic family of type (b) [35, p. 3921; that is

(i) t(K) is a closed sectorial form22.

(iv) The domain Q = Q(t(K)) is independent of K and (u, t(K)U) is

analytic in K for all u E Q. In particular, if H,, + KV is essentially self-adjoint for K > 0, then

the family of closures of the positive K operators has a resolvent analytic continuation to the cut plane.

Before proving this theorem in a series of lemmas, we note that it has the corollary:

COROLLARY 4.20. Let V be as in Theorem 4.1 and let H(K) =

H,, + KV be the closure of the operator sum defined for K > 0. Then H(K) has a resolvent analytic continuation to the cut K-$da?U?.

21 A. Jaffe, private communication and Ref. [33]; see also Ref. [54]. 22 Our definition of sectorial is slightly more general than Kato’s in that we only

require the numerical range to lie in some closed sector of opening angle strictly less than T; he requires the sector to be symmetric about the positive real axis.


Proof. Take V+(Q) = max( V(Q), 0). Condition (b) of Theorem 4.19 is a consequence of the fact that Ha + a-lV- is bounded below for any a. Q.E.D.

Let us turn to the proof of Theorem 4.4. Lemmas 4.21 and 4.22 are both stated supposing conditions (a) and (b) of the theorem hold:

LEMMA 4.21. For any fixed K, H, - KV- has a numerical range in a symmetric translated sector {z 1 1 arg(x - z,,)l < l/2 8) of arbitrarily small opening angle 6.

Proof. Let 11 u II = 1. Then (u, V-u) < a(u, H,u) + b, so

which proves the result since H,, - (Re K) V- is bounded below.

LEMMA 4.22. t(K) = H,, + KV dejked on Q(H,) n Q(V+) is a closed sectorial form if K is not a negative real.

Proof. H, + KV- has numerical range in a sector of opening angle 0 < v - 1 arg K j. Then (H, + K V-) + K V+ is the sum of sectorial forms whose sectors generate a sector of opening angle less than r. Thus t(K) is sectorial. It is not hard to show that if U, is Cauchy in the norms j/l a lilt = I(u, (t(K) + b) + b)u)l, it is Cauchy in the norms

Ill u INHo = (up (8 + 1)~) and Ill u Illv+ = (u, (V+ + lb), so t(K) is closed.

Proof of Theorem 4.19. We have just proven (i), and (ii) is obvious. The final statement of the theorem follows from the theory of perturbations of type (B) [35, pp. 395-3971. Q.E.D.

We have seen for h positive, H0 + XV has a unique vacuum (Theorems 4.5 and 4.14). It thus follows from the general theory of analytic families of operators and Corollary 4.20 that:

PROPOSITION 4.21. Let H, = dr(w), w > cl and V ELP, some p > 2; e-vt E L1 all t > 0. Let G’,, be the vacuum for H, + hV normalized by ]I !S,, /I = 1, fin, 3 0. It has an analytic continuation to a neighborhood of the real axis and the ground state energy

E(h) = GA > H&>l<Q, > .n,> h as a continuation to a neighborhood of the real axis.

Naturally, one is interested in the singularities of E(h) when one


continues it to the cut plane and beyond. Even in the case of one degree of freedom, this is an interesting problem which has recently been studied in the case p2 + x2 + Ax4 by one of us [54] and by Loeffel- Martin [39]. In that case, the principle analyticity results were the following:

(1) E(h) has a continuation without singularities to the cut A plane 1391.

(2) If E(h) d oes not develope natural boundaries, it has a continuation to a domain which is three sheeted about h = 0, with cuts at X # 0 allowed [54].

(3) On this th ree sheeted surface, X = 0 is not an isolated point of analytic or an isolated singularity, but it is a limit point of singularities whose asympototic phase is *3rr/2 [54].“3

(4) For any 0 < 3~12, there is a A with E(h) analytic in (A I 1 A / < A, / arg A / < O> and the Rayleigh-Schrodinger series is asymptoticz4 as h --f 0 uniformly in the sector [54].

We are not able to extend these results to the infinite number of degrees of freedom case 25 but we can prove several weaker results. First, if zero is to be a limit point of singularities (as it almost surely is), they must have asymptotic phase at least n/2, for:

PROPOSITION 4.22. Let H,, , V be as in Proposition 4.21. Let 0 < r/2 be given. Then there is a Q so that E(X) and s2, are analytic in (XjjhIn;iargXj CO> and E(h) is the only eigmvalue of H0 + hV near 0.

Proof. It is enough to show H,, + 1 h 1 eisV converges in norm resolvent sense as j h / -+ 0 for each fixed 1 6 / < 7r/2 with this convergence uniform in 1 0 1 < 0. This follows from Theorem 2.22.

Q.E.D.

Moreover, the Rayleigh-Schrodinger series for E(h) is asymptotic. We first note the lemmas:

LEMMA 4.23. Let w > d; H0 = dF(w) and suppose 0 c [ ,LL 1 < c. Then (H,, - p)-l is a bounded operator of L” into Lp for any 1 < p < co

23 If we are to believe the approximate calculation of C. Bender and T. T. Wu [2, 31 these singularities are square root branch points.

24 It is known to diverge [2, 541. *6 Case (2)-(4) are known for the many (i.e., finite but bigger than 1) degree of

freedom case, but case (1) isn’t even known in the two degree of freedom case.


and the bound is uniform on compact subsets of {p 1 0 < ] p j < c> x

IPI1 <P-4*

Proof. Write LP = Cl @J? where F = {f G Lf’ / J f dp = 01. (Ho - p)-l leaves Cl and 3 invariant so it is sufficient to show (H, - p)-l is bounded on each fi. Without loss of generality, suppose p > 2 since duality can be used for p < 2. Pick T so that ezHo is a contraction of L’L to Lp, and consider how e-(t+T)Ho acts on L*. Since GCS,

Thus

II e--(t+T)H4 IID < eetC II # IID if #GE.

For any TV with Re p < c, J’rLfit e+V# dt exists 2 # ~5 and it is uniformly bounded on the LP unit ball. Thus LP is mapped into itself by (29, - p)-’ in a bounded manner. Q.E.D.

Remark. Let V be real and in L* for some q > 2 with e-lY E Ll for all t. Let H,, = dr(w), o 3 cI. We have already seen that H has a simple L2 eigenvalue at E(V) and a gap of some size e(V) before the next Lpectral point. By an argument identical to the above proof with Lp replaced by {f E LP / (Qn, ,f) = 0}, one can show the LP spectrum of H also has the gap; explicitly

spec(H r Lp) n {z 1 Re z < E(V) + c(V)> = {E(V)}.

In fact, one can prove further that the LP-spectrum is also purely discrete in [E( I’), E(V) + C) and is identical in that interval to the L2-spectrum.

LEMMA 4.24. Let H,, = dr(w); w 3 d, V EL* (some q < oo), e-vt E L1 all t. Suppose 0 < n-12, p, and 0 < E < c are given. Then, thereisaQsothatfor\hI <fi,jargh/ < Oandany~withj~1 = E, (4, + hV - P)-’ is a bounded map of Lp into Lp with bound uniform on the set of p and X in question.

Proof. This follows from Lemma 4.23 and the uniform LP norm resolvant convergence of H, + / h 1 eieV to Ii, as 1 h / -+ 0 with 1 6’ 1 < 0 (Theorem 2.22).

THEOREM 4.25. Let H, = dr(w); w > d; V ELP, all p <co,


real, ervt E L1 all t > 0. Let E(h) and Sz, be the vacuum energy and vacuum vector for Ho + AV. Then,

(a) The Rayleigh-Schriidinger series for E(h) is asymptotic as h 4 0 in any sector {A 1 1 arg h / < 0 < n/2}.

(b) The Rayleigh-Schriidinger series for s2, is asymptotic as h 4 0 in any sector with remainders going to zero in each Lp (p < co).

Proof. Consider first the resolvant (H,, + XV - p)-’ where 1 p / = E and 1 h / < A as in Lemma 4.24. Then applied to vectors in L”, we have, for any fixed N:

(Ho + hV - $1 = i (-A)“[(Ho - p)-1 V]“(H, - p)-’ ?L=O

+ ( -A)N”(Ho + hV - p)-l[ V(H, - p)-y+l. (4.7)

If we apply both sides of this equation to Q, we see that the geometric series for (H,, + [AV - p])-l Q, is asymptotic in LP with remainder terms uniformly bounded in p. The projection onto GA is given by:

PA = -(27ri)-1 J I@l=C

dp(ffo + Al/’ - /J-l, (4.8)

so PJ& has an asymptotic series. Since QA = P,sZ,i(Q, , P,,sZ0)1~2, Q,, has an asymptotic series as does E(X) = (0,) (Ho + AV) Q,)/(Q, , Q,,). But it is well-known that the series obtained by putting (4.7) into (4.8) is just the Rayleigh-Schrijdinger series.

COROLLARY 4.26. Let H,, = dF((h2 + m”)‘/“). Let

v = s

g(x): P($A(x): dx,

where P is a polynomial which is bounded below. Then the following have asymptotic series, asymptotic umyormly in subsectors of / arg h 1 < rr/2:

(a) E,(h), the ground state energy;

(b) (Sz, , +(h,) **a +(h,) 52,), the equal-time vacuum expectation values;

Proof. This follows from the asymptotic results in Theorem 4.25.

Before leaving the question of H, + V with V fixed, we would like to make a few comments on the possibility of these perturbation series


having convergent Pad& approximants [l, 51. For the one degree of freedom anharmonic oscillator Loeffel et al [40] have shown that the diagonal Pad6 approximants for the Rayleigh-Schrodinger series converge to the actual energy levels. The most difficult part of the proof [39], namely that E(h) is analytic in the cut plane is not even known to extend to several degrees of freedom and we have nothing to say about it extending to an infinite number of degrees of freedom case. In the anharmonic oscillator case, one easily established element of the proof is that E(h) is a Herglotz function-what we wish to remark is that this property cannot hold in the ($zfi)Z case, for:

PROPOSITION 4.27. Let E(h) be the ground state energy of HO + Al’ with H, 3 0. If

(a) E(h) is analytic in the cut plane.

(b) V is not bounded below as a form on D(H,,) r\ D(V). Then E(X) f- CA is not a Herglotz function for any C.

Proof. By a well-known property of Herglotz functions, if E(h) + CA were Herglotz, it would be linearly bounded as h--t co along the positive axis; thus E(X) would be linearly bounded, say 1 E(X)1 < DA. Then (#, (HO + hV)#) 3 -DA for any # with /I $11 = 1. This implies ($, V$) > -D contradicting (b). Q.E.D.

In particular, the ground state energy for

is not a Herglotz function, even if it can be proven analytic in the cut plane. It is rather simple to understand heuristically why this happens. If we take p2 + x2 + Ax4 and partially “Wick-order” x4 by subtracting c = (J&, , x452,), the new energy E(A) = E(h) - CA is no longer Herglotz either. Of course, E(h) + h c is. In the field theory case, Wick ordering involves infinite subtractions, so no finite constant can restore the Herglotz nature. One is tempted to suggest that the Pade approximants to E(X) still might converge, with the Rayleigh-Schrijdinger coefficients, a nob still a series of Steiljes [l] for n 3 2, i.e.,

% = (- l)“r+l so xn dp might hold for n > 2. But, unlike the one degree of freedom case J’r x dp would be infinite.26 Because of this loss of the Herglotz property, it will be hard to prove the discontinuity across the cut is positive even if cut plane analyticity is established.

28 The divergence coming from x = 0.

176 SIRION AND HOEGH-KROHN

We would like to end this section with a brief discussion of how coupling constant analyticity of the physical vacuum in a neighborhood of the real axis might hold or fail to hold. According to the discussion of Wightman [63], there are two ways in which singularities of o~J’~r’ might develop on the real axis as g -+ lz7:

(a) For g E L1 n L2, there are complex singularities in s2, which as g --t 1 “pinch” the real axis.

(b) There is some nonuniformity in the approach to the limit o++Whasg-+l.

We should like to discuss how this nonuniformity (in (b)) would have to manifest itself. Suppose there is some neighborhood N of h, > 0 so that for g, in a sequence g, -+ 1, sZ,,% , the ground state of 4, + X J-g+> :P(4(4): d X, is analytic in N (i.e., suppose (a) does not occur). We can, without loss, normalize L? so ] a(Q)” d&I) = 1. We emphasize that since s;! is complex for X non-real, /I !Z //a > 1 for X nonreal. Any nonuniformity must manifest itself by // L’ II2 + co for:

THEOREM 4.27. Suppose Qn,, is analytic for X E N for some g, with g, -+ 1. Suppose also, that /I a,,, /I2 is bounded as n -+ GO uniformly on compacts of N. Then there is a subsequence g,ci) so that for any h E N,

converges as i -+ CO in the weak star topology to a limiting functional We analytic in N.

Proof. By a result of Jaffe and Glimm [22], we can pick a subsequence with w, I convergent for some fixed real h, . By a standard argument, we can’iet a subsequence converging for h any real rational in N. Since // Q,,, lj2 is bounded on compacts, j UJ~,~(A)~ < 11 .n,,, II2 11 A I/ is also bounded on compacts. The theorem now follows from the Vitali convergence theorem. Q.E.D.

5. APPLICATIONS TO NONRELATAVISTIC QUANTUM MECHANICS

We would like to briefly discuss two results in nonrelatavistic quantum mechanics which the methods of this paper enable us to prove. First, Theorem 2.25 implies:

27 In analogy with thermodynamics, one is tempted to call these singularities “phase transitions”; they might occur, for example, at values of the coupling constant at which bound states appear in the theory.


THEOREM 5.1. Let V be a measurable function on R” so that:

(a) V(q) 3 aq2 - b for some a and b; a > 0;

W S I VWkxp(--a1%2)1 d”q < CO.

Then -A + V is essentially self-adjoint (as an operator on L2(Rn, dnq)) on D(-A) n D(V).

Remark. This should be compared with the Jaffe-Carleman theorem 19, 33, 621 which requires smoothness for V but in place of (a) and (b) only demands V > -b.

Proof. Write -A + V = (-A + aq2) + (V - aq2). -A + aq2 is unitary equivalent to the generator of a hypercontractive semigroup. Case (a) tells us V - aq2 is bounded below and (b) tells us V is in L2 of the relevant measure space. Q.E.D.

Our uniqueness of the vacuum proof (Theorem 4.12) immediately implies the nondegeneracy and nodelessness of the ground state of n-body Bose systems and n-body systems without statistics.

THEOREM 5.2. Let H, = ~:i~j=, aiiFt * pj be the Hamiltonian of a free n-body system on R3n, i.e., aii is a positive de$nite matrix. Let V = Cr<“,j=, Vij(rij) + Cy=, Vi(rJ. Put V’“‘(q) = V(q) if / V(q)1 < n and Vfn)(q) = 0, otherwise. Suppose a Hamiltonian H can be defined so that

(a) lim,,, (H,, + V(%) + X)-l = (H + h)-l in norm.

(b) limn+m (H - V(n) + X)-l = (H,, + X)-l in norm.

Then H, + V has a unique a.e. positive ground state if it has a ground state. This holds true if the Hamiltonian is symmetric under certain particle interchanges and we restrict ourselves to the symmetric (Bose) subspace.

Remarks. (1) Cases (a) and (b) hold, in particular, if each Vi , Vii E L2 + L” or more generally if Vi, Vij E R + L” [55].

(2) This should be compared with the usual proof [12] which depends critically on smoothness assumptions on any nodes of wavefunctions.

Notes added to proofs.

(1) During the typing of this manuscript Segal has announced a result of the form of Corollary 4.14 in his talk before the International Congress of Mathemati- cians.”


(2) Since the appearance of the original manuscript, Proposition 4.22 has been improved to 8 < x with consequences for summability of the perturbation series for E(h); see B. Simon, Bore1 summability of the ground-state energy in spatially cutoff (#J”)~, Phys. Rev. Lett. 25 (1970), 1583-1586, and L. Rosen and B. Simon, Trans. A.M.S. (to appear).

(3) Recently, L. Gross has shown that the existence of a ground state (part of Theorem 4.5) follows from abstract hypercontractivity results alone, see L. Gross, “A noncommutative extension of the Perron-Frobenius theorem,” Bull. A.M.S. 77 (1971), 343-347.

(4) The functions F used as smearing functions for coherent vectors (Egn 3-66) should be functions with 3 in L” rather than only smooth functions.

(5) The one particle space in 3C should be those h E L’(R, dJz) with & real- valued.

(6) In the proof of Prop. 2.1 we did not show that eetHo was a contraction on complex valued functions. This follows from the fact that it is a contraction on real valued functions and

%la” ,- b2 = c 1 a cos 8 + b sin 8 ! d6’ for suitable c. II

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